Obama met in November with Boehner, as well as Senate Majority Leader Harry Reid and House Minority Leader Nancy Pelosi. The president spoke Friday by telephone with Reid and in person with Pelosi.
Obama has been pushing higher tax rates on the wealthiest Americans as one way to reduce the deficit - a position Boehner and other House Republicans have been steadfastly against. Republicans are demanding steeper cuts in costly government entitlement programs like Medicare and Social Security.
One GOP senator said Sunday that Senate Republicans would probably agree to higher tax rates on the wealthiest Americans if it meant getting a chance to overhaul entitlement programs.
The comments by Bob Corker of Tennessee - a fiscal conservative who has been gaining stature in the Senate as a pragmatic deal broker - puts new pressure on Boehner and other Republican leaders to rethink their long-held assertion that even the very rich shouldn't see their rates go up next year. GOP leaders have argued that the revenue gained by hiking the top two tax rates would be trivial to the deficit and that any tax hike hurts job creation.
But Corker said that insisting on a red line, especially since Obama won re-election after campaigning on raising tax rates on the wealthy, might not be wise.
"There is a growing group of folks looking at this and realizing that we don't have a lot of cards as it relates to the tax issue before year end," Corker told Fox News Sunday.
If Republicans agree to Obama's plan to increase rates on the top 2 percent of Americans, Corker added, "the focus then shifts to entitlements, and maybe it puts us in a place where we actually can do something that really saves the nation."
Besides getting tax hikes through the Republican-dominated House, Corker's proposal faces another hurdle: Democrats haven't been receptive to GOP proposals on the entitlement programs. Senate Democratic Whip Dick Durbin (D., Ill.) was skeptical Sunday about proposals to increase the eligibility age for Medicare from 65 to 67. He said he couldn't see Congress addressing the complicated issue of Medicare overhaul in the three weeks remaining before 2013.
"I just don't think we can do it in a matter of days here before the end of the year," Durbin said. "We need to address that in a thoughtful way through the committee structure after the first of the year."
And hard-line fiscal conservatives in the House are holding fast to their position.
"No Republican wants to vote for a tax-rate increase," said Rep. Jeb Hensarling (R., Texas), chairman of the House Republican Conference.
Added Rep. Marsha Blackburn (R., Tenn.): "I'm not sure there is support for the rate hikes. There is support for revenue by cleaning up the code."
Still, at least one House Republican has said there was another way. Rep. Tom Cole, of Oklahoma, has said Obama and Boehner should agree not to raise tax rates on the majority of Americans and negotiate the rates for top earners later. Cole said Sunday that most House Republicans would vote for that approach because it didn't include a rate hike.
"You know, it's not waving a white flag to recognize political reality," Cole said.
Sen. Tom Coburn (R., Okla.) already has said he could support higher tax rates on upper incomes as part of a comprehensive plan to cut the deficit.
When asked Sunday what it would take to sign on to a tax-rate increase, Coburn echoed Corker's comments by responding, "Significant entitlement reform." He quickly added, however, that he has estimated that such a tax-rate increase would affect only about 7 percent of the deficit.
"Will I accept a tax increase as a part of a deal to actually solve our problems? Yes," Coburn said. "But the president's negotiating with the wrong people. He needs to be negotiating with our bondholders in China, because if we don't put a credible plan on the discussion, ultimately, we all lose."
Obama's plan would raise $1.6 trillion in revenue over 10 years, partly by letting the decade-old tax cuts on the country's highest earners expire at the end of the year. He would continue those Bush-era tax cuts for everyone except individuals earning more than $200,000 and couples making more than $250,000. The highest rates on top-paid Americans would rise from 33 percent and 35 percent to 36 percent and 39.6 percent.
Boehner has offered $800 billion in new revenue to be raised by reducing or eliminating unspecified tax breaks on upper-income people. The Republican plan would cut spending $1.4 trillion, including by trimming annual increases in Social Security payments and raising the eligibility age for Medicare.
Hensarling and Coburn spoke on ABC's This Week. Blackburn and Cole spoke on CNN's State of the Union. Durbin spoke on NBC's Meet the Press.