In the Republicans' weekly address, Sen. Roy Blunt of Missouri cited a readiness to compromise. "Divided government is a good time to solve hard problems and in the next few days, leaders in Washington have an important responsibility to work together and do just that," he said.
Even so, there was no guarantee of success, and a dispute over the federal tax on large estates emerged as yet another key sticking point alongside personal income tax rates.
In a blunt challenge to Republicans, Obama said that barring a bipartisan agreement, he expected both houses to vote on his own proposal to block tax increases on all but the wealthy and simultaneously preserve expiring unemployment benefits.
Political calculations mattered as much as deep-seated differences over the issues, as divided government struggled with its first big challenge since the November elections.
Speaker John Boehner remained at arms-length, juggling a desire to avoid the fiscal cliff with his goal of winning another term as speaker when a new Congress convenes next Thursday. Any compromise legislation is certain to include higher tax rates on the wealthy, and the House GOP rank and file rejected the idea when he presented it to them as part of a final attempt to strike a more sweeping agreement with Obama.
Lawmakers have until the new Congress convenes to pass any compromise, and even the calendar mattered. Democrats said they had been told House Republicans might reject a deal until after Jan. 1, to avoid a vote to raise taxes before they had technically gone up and then vote to cut taxes after they had risen.
Nor was any taxpayer likely to feel any adverse impact if legislation is signed and passed into law in the first two or three days of 2013 instead of the final hours of 2012.
Gone was the talk of a grand bargain of spending cuts and additional tax revenue in which the two parties would agree to slash deficits by trillions of dollars over a decade.
Now negotiators had a more cramped goal of preventing additional damage to the economy in the form of higher taxes across the board with some families facing increases measured in the thousands of dollars as well as cuts aimed at the Pentagon and hundreds of domestic programs.
Republicans said they were willing to bow to Obama's call for higher taxes on the wealthy as part of a deal to prevent them from rising on those less well-off.
Democrats said Obama was sticking to his campaign call for tax increases above $250,000 in annual income, even though he said in recent negotiations he said he could accept $400,000. There was no evidence of agreement even at the higher level.
There were indications from Republicans that estate taxes might hold more significance for them than the possibility of higher rates on income.
One senior Republican, Sen. Jon Kyl of Arizona, said late Friday he was "totally dead set" against Obama's estate tax proposal, and as if to reinforce the point, Blunt mentioned the issue before any other in his broadcast remarks. "Small businesses and farm families don't know how to deal with the unfair death taxa tax that the president and congressional leaders have threatened to expand to include even more family farms and even more small businesses," he said.
Several officials said Republicans want to leave the tax at 35 percent after exempting the first $5 million in estate value. Officials said the White House wants a 45 percent tax after a $3.5 million exemption. Without any action by Congress, it would climb to a 55 percent tax after a $1 million exemption on Jan. 1.
Democrats stressed their unwillingness to make concessions on both income taxes and the estate tax, and said they hoped Republicans would choose which mattered more to them.
Officials said any compromise was likely to ease the impact of the alternative minimum tax, originally designed to make sure that millionaires did not escape taxation. If left unchanged, it could hit an estimated 28 million households for the first time in 2013, with an average increase of more than $3,000.
Taxes on dividends and capital gains are also involved in the talks, as well as a series of breaks for businesses and others due to expire at the first of the year.
Obama and congressional Democrats are insisting on an extension of long-term unemployment benefits that are expiring for about 2 million jobless individuals.
Leaders in both parties also hope to prevent a 27 percent fee cut from taking effect on Jan. 1 for doctors who treat Medicare patients.
There was also discussion of a short-term extension of expiring farm programs, in part to prevent a spike in milk prices at the first of the year. It wasn't clear if that was a parallel effort to the cliff talks or had become wrapped into them.
Across-the-board spending cuts that comprise part of the cliff were a different matter.
Republicans say Boehner will insist that they will begin to take effect unless negotiators agreed to offset them with specified savings elsewhere.
That would set the stage for the next round of brinkmanship a struggle over Republican calls for savings from Medicare, Medicaid and other federal benefit programs.
The Treasury's ability to borrow is expected to expire in late winter or early spring, and without an increase in the $16.4 trillion limit, the government would face its first-ever default. Republicans have said they will use administration requests for an extension as leverage to win cuts in spending.
Ironically, it was just such a maneuver more than a year ago that set the stage for the current crisis talks over the fiscal cliff. Over the fiscal cliff: Soft or hard landing
Efforts to save the nation from going over a year-end "fiscal cliff" were still in disarray as lawmakers returned to the Capitol to confront the tax-and-spend crisis. A tone-setting quotation was Democratic Sen. Harry Reid's assertion that the House under Republican Speaker John Boehner had been "operating with a dictatorship."
President Barack Obama flew back to Washington from Hawaii after telephoning congressional leaders from his Christmas vacation perch. Once back, he set up a meeting with leaders of both parties at the White House late Friday to make a fresh attempt to find a solution before Monday night's deadline.
A look at why it's so hard for Republicans and Democrats to compromise on urgent matters of taxes and spending, and what happens if they fail to meet their deadline:
NEW YEAR'S HEADACHE
Partly by fate, partly by design, some scary fiscal forces come together at the start of 2013 unless Congress and Obama act to stop them. They include:
Some $536 billion in tax increases, touching nearly all Americans, because various federal tax cuts and breaks expire at year's end.
About $110 billion in spending cuts divided equally between the military and most other federal departments. That's about 8 percent of their annual budgets, 9 percent for the Pentagon.
Hitting the national economy with that double whammy of tax increases and spending cuts is what's called going over the "fiscal cliff." If allowed to unfold over 2013, it would lead to recession, a big jump in unemployment and financial market turmoil, economists predict.
WHAT IF THEY MISS THE DEADLINE?
If New Year's Day arrives without a deal, the nation shouldn't plunge onto the shoals of recession immediately. There still might be time to engineer a soft landing.
So long as lawmakers and the president appear to be working toward agreement, the tax hikes and spending cuts could mostly be held at bay for a few weeks. Then they could be repealed retroactively once a deal was reached.
The big wild card is the stock market and the nation's financial confidence: Would traders start to panic if Washington appeared unable to reach accord? Would worried consumers and businesses sharply reduce their spending? In what could be a preview, stock prices in the U.S. and Europe dropped Friday on waning hopes that Obama and key lawmakers would reach an 11th-hour compromise.
Federal Reserve Chairman Ben Bernanke has warned lawmakers that the economy is already suffering from the uncertainty and they shouldn't risk making it worse by blowing past their deadline.
WHAT IF THEY NEVER AGREE?
If negotiations between Obama and Congress collapse completely, 2013 looks like a rocky year.
Taxes would jump $2,400 on average for families with incomes of $50,000 to $75,000, according to a study by the nonpartisan Tax Policy Center. Because consumers would get less of their paychecks to spend, businesses and jobs would suffer.
At the same time, Americans would feel cuts in government services; some federal workers would be furloughed or laid off and companies would lose government business. The nation would lose up to 3.4 million jobs, the Congressional Budget Office predicts.
"The consequences of that would be felt by everybody," Bernanke says.
Much of the disagreement surrounds the George W. Bush-era income tax cuts, and whether those rates should be allowed to rise for the nation's wealthiest taxpayers. Both political parties say they want to protect the middle class from tax increases.
Several tax breaks begun in 2009 to stimulate the economy by aiding low- and middle-income families are also set to expire Jan. 1. The alternative minimum tax would expand to catch 28 million more taxpayers, with an average increase of $3,700 a year. Taxes on investments would rise, too. More deaths would be covered by the federal estate tax, and the rate climbs from 35 percent to 55 percent. Some corporate tax breaks would end.
The temporary Social Security payroll tax cut also is due to expire. That tax break for most Americans seems likely to end even if a fiscal cliff deal is reached, now that Obama has backed down from his call to prolong it as an economic stimulus.
If the nation goes over the fiscal cliff, budget cuts of 8 percent or 9 percent would hit most of the federal government, touching all sorts of things from agriculture to law enforcement and the military to weather forecasting. A few areas, such as Social Security benefits, Veterans Affairs and some programs for the poor, are exempt.
THERE'S MORE AT STAKE
All sorts of stuff could get wrapped up in the fiscal cliff deal-making. A sampling:
Some 2 million jobless Americans may lose their federal unemployment aid. Obama wants to continue the benefits extension as part of the deal; Republicans say it's too costly.
Social Security recipients might see their checks grow more slowly. As part of a possible deal, Obama and Republican leaders want to change the way cost-of-living adjustments are calculated, which would mean smaller checks over the years for retirees who get Social Security, veterans' benefits or government pensions.
The price of milk could double. If Congress doesn't provide a fix for expiring dairy price supports before Jan. 1, milk-drinking families could feel the pinch. One scenario is to attach a farm bill extension to the fiscal cliff legislation if a compromise is reached in time.
Millions of taxpayers who want to file their 2012 returns before mid-March will be held up while they wait to see if Congress comes through with a deal to stop the alternative minimum tax from hitting more people.
CALL THE WHOLE THING OFF?
In theory, Congress and Obama could just say no to the fiscal cliff, by extending all the tax cuts and overturning the automatic spending reductions in current law. But both Republicans and Democrats agree it's time to take steps to put the nation on a path away from a future of crippling debt.
Indeed, the automatic spending cuts set for January were created as a last-ditch effort to force Congress to deal with the debt problem.
If Washington bypassed the fiscal cliff, the next crisis would be just around the corner, in late February or early March, when the government reaches a $16.4 trillion ceiling on the amount of money it can borrow.
Boehner says Republicans won't go along with raising the limit on government borrowing unless the increase is matched by spending cuts to help attack the long-term debt problem. Failing to raise the debt ceiling could lead to a first-ever U.S. default that would roil the financial markets and shake worldwide confidence in the United States.
To avoid that scenario, Obama and Boehner are trying to wrap a debt limit agreement into the fiscal cliff negotiations.
SO WHAT'S THE HOLDUP?
They're at loggerheads over some big questions.
Obama says any deal must include higher taxes for the wealthiest Americans. Many House Republicans oppose raising anyone's tax rates. Boehner tried to get the House to vote for higher taxes only on incomes above $1 million but dropped the effort when it became clear he didn't have the votes.
Republicans also insist on deeper spending cuts than Democrats want to make. And they want to bring the nation's long-term debt under control by significantly curtailing the growth of Medicare, Medicaid and Social Security changes that many Democrats oppose.
Obama, meanwhile, wants more temporary economic "stimulus" spending to help speed up a sluggish recovery. Republicans say the nation can't afford it.
IT'S NOT JUST WASHINGTON
Seems like they could just make nice, shake hands and split their differences, right?
But there's a reason neither side wants to give ground. The two parties represent a divided and inconsistent America. True, Obama just won re-election. But voters also chose a Republican majority in the House.
Republicans and Democrats alike say they are doing what the voters back home want.
Neither side has a clear advantage in public opinion. In an Associated Press-GfK poll, 43 percent said they trust the Democrats more to manage the federal budget deficit and 40 percent preferred the Republicans. There's a similar split on who's more trusted with taxes.
About half of Americans support higher taxes for the wealthy, the poll says, and about 10 percent want tax increases all around. Still, almost half say cutting government services, not raising taxes, should be the main focus of lawmakers as they try to balance the budget.
When asked about specific budget cuts being discussed in Washington, few Americans express support for them.
Time for deal-making is short, thanks to the holiday and congressional calendars. Some key dates for averting the fiscal cliff:
Lawmakers didn't begin returning to the Capitol until Thursday, leaving less than a week to vote on a compromise before year's end.
Obama returned Thursday from his Christmas vacation in Hawaii. The president asked congressional leaders to the White House Friday to try to resolve the fiscal cliff.
If lawmakers reach Dec. 31 without a deal, some economists worry that the financial markets might swoon.
The current Congress is in session only through noon Eastern time on Jan. 3. After that, a newly elected Congress with 13 new senators and 82 new House members would inherit the problem.
Associated Press writers Jim Kuhnhenn, Alan Fram and Andrew Taylor and Director of Polling Jennifer Agiesta contributed to this report.