Treasury Secretary Jacob Lew has said the deadline for Congress to act is Oct., 17, setting that as the day the government will exhaust its ability to borrow funds and will have to rely day-to-day on tax and other receipts to pay its bills.
Some Republicans have downplayed the significance of the deadline, saying that even then, the United States would be able to pay China and other holders of U.S. debt.
But Obama said they were badly misguided, warning that default would harm the economy, cause retirement accounts to shrivel and houses to lose value.
Other Republicans have made it clear in recent days they agree with the threat posed by default and are determined to prevent it.
"I suspect we can work out a mechanism to raise the debt ceiling while a negotiation is underway," said Rep. Tom Cole, an Oklahoma Republican, who is close to Speaker John Boehner.
"I want to have a conversation," Boehner told reporters. "I'm not drawing lines in the sand. It is time for us to just sit down and resolve our differences."
In the Senate, Majority Leader Harry Reid readied legislation to raise the debt limit by roughly $1 trillion, enough to prevent a recurrence of the current showdown until after the 2014 elections.
It was unclear whether Senate Republicans would slow progress of the bill, which was shorn of all items that many GOP lawmakers favor to reduce deficits or delay the health overhaul, which takes effect more fully on Jan. 1.
Inside the Capitol, the threat of a default overshadowed the continuing partial government shutdown, in its eighth day with little or no talk of an immediate end. An estimated 450,000 federal workers are idled at agencies responsible for items as diverse as food inspection and national parks, although all employees are eventually expected to receive full back pay.
In the House, majority Republicans announced plans to pass legislation reopening Head Start, the pre-school program for disadvantaged children. It is the latest in a string of bills to end the shutdown in one corner of government or another in hopes of forcing Democrats to abandon their own demands for a full reopening of the federal establishment.
Republicans also announced they would vote to make sure federal workers on the job don't miss their next regularly scheduled paycheck on Oct. 15.
In a potentially more significant political development, a third vote was expected on legislation to create a House-Senate working group on deficit reduction and economic growth. The 20-member panel would be empowered to recommend steps to raise the debt limit and reduce spending, including in so-called direct spending programs a definition that appears broad enough to encompass benefit programs like Medicare and Social Security as well as the health care law that Republicans oppose.
The measure does not contain any provision to end the shutdown or raise the debt limit, although it could be amended to include them at a later date if a compromise emerges.
The shutdown began more than a week ago after Obama and Senate Democrats rejected Republican demands to defund "Obamacare," then to delay it, and finally to force a one-year delay in the requirement for individuals to purchase health care coverage or face a financial penalty.
It was not a course Boehner and the leadership had recommended preferring a less confrontational approach and hoping to defer a showdown for the debt limit. Their hand was forced by a strategy advanced by Texas Sen. Ted Cruz and tea party-aligned House members determined to eradicate the health care law before it fully took root.
That portion of the strategy was doomed to failure, since money for the health care program was never cut off.
With the government partially shut down, Boehner and the GOP leadership decided to allow the closure showdown to merge with one over the debt limit.
Officials who attended a closed-door meeting of the Republicans' House rank and file said Boehner had told lawmakers they were in the midst of a tough fight and Obama and Reid were trying to "annihilate" them.
House Majority Leader Eric Cantor said the current showdown isn't about Obamacare or the government shutdown, but about the constitutional balance of power between the president and Congress.
Democrats ridiculed the legislation to create a working group of lawmakers, likening it to a supercommittee that failed two years ago to reach a compromise on measures to cut future spending.
At the same time, there were signs of early maneuvering for political position in case the group is established.
"They claim they want to talk about deficit reduction, but their bill immediately rules out talking about closing tax loopholes to help get our fiscal house in order," said Rep. Chris Van Hollen, D-Md..
"It's just more of the same from Speaker Boehner and his tea party caucus."
Associated Press writers Donna Cassata, Andrew Taylor, Alan Fram, Martin Crutsinger, Julie Pace and Chuck Babington contributed to this story.