Republicans seeking to shrink the size of government oppose increasing any tax rates, arguing that Obama's plan would hinder job growth because some small business owners who file personal returns would pay higher taxes under it.
While aides on both sides have been talking, no follow-up meeting between Obama and congressional leaders has been scheduled after their initial post-election discussion on Nov. 16.
Instead, Obama met Tuesday with small business owners, the first in a series of events this week intended to highlight his push for raising taxes on the wealthiest 2 percent of Americans while maintaining current rates for everyone else.
After the White House gathering, three of the 15 small business owners who took part said they wanted certainty about their own tax situations while preventing middle-class Americans from paying higher taxes.
Andra Rush, who founded Rush Trucking of Wayne, Michigan, said her message to Obama was that failure to extend the tax cuts to the middle class could stall what she called new economic momentum in the country.
"I would have higher tax rates," Rush conceded, adding that it was more important for "ordinary Americans" to have more money to spend instead of paying it in taxes if everyone's rates go up.
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 percent of Americans would blame Republicans for failing to avoid the fiscal cliff, while 34 percent 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.
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 percent-44 percent.
Without a deal, tax cuts from 2001 and 2003 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."