Defense Secretary Leon Panetta also wants to retire next year.
"The first thing is to try to find a way out of the box we're in with regards to the fiscal cliff," said Tom Daschle, the former Senate majority leader who is close to Obama. "When the new Congress convenes they'll begin the nominating process for what I expect will be a good number of vacancies."
Obama privately delved into both issues Thursday, his first full day back in Washington following his re-election on Tuesday. The president and his team were also assessing how congressional Republicans were positioning themselves following the election before saying much publicly about his second term.
The president will make his first postelection comments on the economy and the fiscal cliff Friday at the White House.
Obama aides want to avoid what they believe was an overreach by President George W. Bush, who declared after narrowly winning re-election that he had "political capital" and intended to spend it.
One of Bush's first moves was to push to privatize Social Security, a plan that was roundly rejected by Congress and the public.
The White House believes Obama has a clear mandate on one key issue: raising taxes on families making more than $250,000 a year. Obama senior adviser David Plouffe said voters "clearly chose the president's view of making sure the wealthiest Americans are asked to do a little bit more" to help shrink the federal deficit.
The president has long advocated allowing tax cuts first passed by Bush to expire for upper income earners. But he gave in to Republican demands in 2010 and allowed the cuts to continue, angering many Democrats.
Republican House Speaker John Boehner said Wednesday that he wanted to compromise with the re-elected president.
And he said the House would be willing to accept higher tax revenue under the right conditions as part of a more sweeping attempt to reduce deficits.
The White House wants consistency in its "fiscal cliff" negotiating team, meaning Geithner is likely to put off his departure from Treasury until Obama and lawmakers can reach some agreement.
White House chief of staff Jack Lew is seen as a leading candidate to replace Geithner. Lew is well-respected in Washington by both parties.