Boehner and other influential GOP figures have declared their willingness to consider other ways to boost tax revenue as part of a broader deal that includes entitlement reforms and spending cuts.
That position undermines the no-tax-increase pledge championed by anti-tax crusader Grover Norquist, which Democrats consider to be a major impediment to a deficit reduction deal.
Some Republicans -- including conservative Sens. John McCain of Arizona, Bob Corker of Tennessee, Saxby Chambliss of Georgia, Tom Coburn of Oklahoma and Lindsay Graham of South Carolina as well as Rep. Peter King of New York and Scott Rigell of Virginia -- have dropped their adherence to Norquist's pledge.
Norquist responded harshly Monday night, telling CNN that those denouncing the pledge were breaking their commitment to voters. He targeted King for saying a pledge signed years ago no longer applied.
King "tried to weasel out" of the pledge, adding: "I hope his wife understands commitments last a little longer than two years or something."
According to Norquist, King knew when he signed the pledge that it applied to "as long as you're in Congress."
"It's only as long as you're in the House or the Senate. If he stayed too long, that's his problem," Norquist added.
To CNN Chief Political Analyst Gloria Borger, the softening tone by some in the GOP was explained by new poll numbers that showed 45% of Americans would blame Republicans for failing to avoid the fiscal cliff, while 34% would blame Obama.
Republicans insist Democrats must agree to cut discretionary spending and make significant reforms to Medicare and Social Security as part of a deficit reduction deal.
However, organized labor and other elements of the Democratic base oppose any major reforms to the popular entitlement programs. While some Democratic legislators express willingness to reform Medicare and Medicaid, they reject making Social Security reform part of the fiscal cliff negotiations, saying it is self-funded and therefore doesn't add to the deficit.
Obama spoke by phone over the weekend with Boehner and Senate Democratic leader Harry Reid, Carney said Monday, and he rejected a reporter's assertion Tuesday that no progress was taking place in efforts to avoid the fiscal cliff.
On Capitol Hill, a Senate Democratic leadership aide said Monday that behind-the-scenes talks have "haven't been tremendously productive."
Negotiations were not at a point "where people are huddled in a conference room with spread sheets" but are instead at the early "dancing stage," the Democratic aide said.
Congressional Republican leadership aides declined to characterize the state of the talks, though one noted the fact that discussions continued could be interpreted as progress.
The CNN/ORC International poll released Monday also showed that a solid majority of respondents -- two thirds -- supports the Democratic stance that any agreement should include a mix of spending cuts and tax increases. Of that total, Republicans favor such an approach by 52%-44%.
Without a deal, tax cuts from 2001 and 2003 -- when George W. Bush was president -- will expire, raising rates for everyone starting in January. In addition, spending cuts would reduce spending on the military, national parks, the Federal Aviation Administration and other government services.
However, the government and Congress still would have time to prevent draconian effects from the fiscal cliff when a new Congress convenes in January.
William Galston, a senior fellow in governance studies at the Brookings Institution, called that a form of brinksmanship best avoided.
"To be sure, no one believes that non-agreement by December 31 would be the end of the story. After a period of finger-pointing, discussions would resume," he wrote last week in a New Republic opinion piece. "But equally, no one knows how the failure to reach agreement before the end of 2012 would affect the dynamics of the negotiations."
In addition, "we can be reasonably sure ... that national and global markets would react adversely and that businesses, which are already retreating from planned investments in new plant and equipment, would become even more uncertain and risk-averse."