But Republicans also know they have a problem: many liberal Democrats are more than willing to return to the Clinton-era tax code, and to allow across-the-board spending cuts to take effect, which disproportionately affect the military, rather than compromise too much with Republicans after the strong Democratic showing in the elections.

“It’s a terrible position because by default, Democrats get what they want,” said Representative James Lankford, Republican of Oklahoma, who admitted his party is boxed in.

In the debt ceiling fight, the tables might turn. Many conservative Republicans say they are willing to let the nation default if Congress refuses to cut spending. If the tax rate fight is resolved by the time the debt limit increase is needed, Democrats will find themselves without the leverage they now have with the expiration of the lower tax rates a certainty unless Congress acts affirmatively.

That is why in his opening bid to end the fiscal standoff, Mr. Obama proposed a permanent policy change to let the president raise the nation’s borrowing limit on his own — and why Senator Mitch McConnell of Kentucky, the Republican leader, reportedly laughed out loud at the idea.

Popular opinion seems to be running against the Republicans. Most Americans support higher taxes on the rich, and a poll by The Washington Post and the Pew Research Center released Tuesday found that 53 percent say Republicans in Congress would and should be blamed if Washington fails to reach a deal before January. Just 27 percent said Mr. Obama would deserve more blame, while 12 percent say both sides would share the blame equally.

Most Republicans do not want tax rates to rise on anyone, and such national polls have little sway over House members in districts drawn to favor one party over the other. In November, 204 Republicans — 88 percent of the House Republican Conference — won at least 55 percent of the vote, according to David Wasserman, a House analyst at The Cook Political Report. Thirty-five of them won at least 70 percent.

Speaker Boehner took as much fire from conservatives as from Democrats after proposing a deficit-reduction plan that would raise $800 billion in tax revenue over 10 years. Conservative advocacy groups and conservatives on Capitol Hill were united in their condemnation.