Mr. Obama has signaled a willingness to reduce cost-of-living increases for Social Security by using a less-generous measure of inflation. He has indicated openness to imposing means-testing on Medicare beneficiaries so that high-income retirees would pay more for their medical care, and he has put on the table $400 billion of cuts in Medicare over the next decade, mostly through reductions in payments to health care providers.
Republicans say they are looking for more, including two elements they discussed with the White House during failed talks in 2011: raising the eligibility age for Medicare and cutting federal costs for Medicaid.
White House officials said those proposals were deeply flawed as a matter of policy and that they did not intend to submit anymore offers until Republicans express some willingness to make tax revenue part of the equation.
But Mr. Cantor raised an idea last month that had been endorsed by the Bowles-Simpson deficit-reduction commission: merging Medicare's hospital and doctor coverage into one program in a way that could generate savings. Some Democrats also see that kind of proposal — in which Medicare is left fundamentally intact but is overhauled to become more efficient and which could potentially charge the affluent elderly more — as the basis for negotiation.
With all the familiar obstacles looming, however, Democrats are increasingly looking at the previous round of negotiations — at the end of 2012, as the Bush-era tax cuts were scheduled to expire — and concluding that Mr. Obama flinched, leaving tax revenue on the table that would have ended the budget standoff on more favorable terms to Democrats.
"There's a lot of Monday morning quarterbacking," said Representative Chris Van Hollen of Maryland, the ranking Democrat on the House Budget Committee.
The second-guessing extends to virtually every aspect of the deal: its failure to postpone the automatic budget cuts for more than two months, its failure to raise the federal debt limit, its yield of $600 billion in new tax revenue over 10 years out of $4 trillion of new taxes that would have taken effect had the Bush tax cuts been allowed to expire.
Senator Amy Klobuchar, Democrat of Minnesota, said the White House should have negotiated after the election with the bipartisan group of senators that he is courting now, rather than resuming the talks with House Speaker John A. Boehner that failed in 2011. In the agreement that concluded that 2011 standoff, said John Podesta, a former White House chief of staff for President Bill Clinton, the administration committed a "fundamental miscalculation" by believing Republican opposition to automatic Pentagon cuts would compel them to later accept tax increases.
Other Democratic critics say the president should have ratcheted up pressure on Republicans by allowing all the Bush-era tax cuts to expire, even at the risk of saddling middle class families with tax increases and inflicting harm on the economy.
"Thinking they'd have a second bite of the apple was a real mistake," said Robert D. Reischauer, a former director of the Congressional Budget Office, since any tax increase would inflame passions among conservatives and limit Mr. Boehner's maneuvering room.
White House aides said the deal was a good one. Not only did they crack Republican resistance to income tax rate increases, they said, but they did it without committing to any major cuts to Medicare and Social Security beneficiaries.
They said that the president was not willing to risk the economy's health by forcing showdowns when he could get much of what he wants through negotiation, and that the president had been able to make progress on changes to immigration policy and gun control thanks in part to the deal.
Moreover, White House officials said their negotiating position had been undercut by calls from prominent Democrats like Representative Nancy Pelosi, the House minority leader, and Senator Charles E. Schumer, the No. 3 Democrat in the Senate, for tax increases that would affect only families with incomes of at least $500,000, rather than Mr. Obama's preferred $250,000 threshold. (The deal raised the top rate on income over $450,000 for couples and $400,000 for single people). And had they failed to reach a quick deal, White House officials said, they could have lost the opportunity for progress on the rest of the president's second-term agenda.
"If we had followed the advice of some, the most likely scenario is that today everyone's taxes would still be raised," Mr. Pfeiffer said. "The sequester would have been in effect for three months; we would be a month or so away from the debt ceiling; and all the other Democratic priorities would be nowhere."
Some former members of Mr. Obama's economic team said the White House could have gotten more. Jared Bernstein, a former adviser to Vice President Joseph R. Biden Jr., faulted the administration for agreeing to extend the bulk of the Bush-era tax cuts rather than raising more tax revenue that could be used to pay for other priorities.
Peter R. Orszag, who was Mr. Obama's first budget director, said, "By making the middle class tax cuts permanent, we've unfortunately locked into a revenue base that is inadequate."