Obama argued Wednesday that settling the tax question for middle-class families would clear the way for the broader agreement everyone wants.
"We can do it in a balanced a fair way, but our first job is to make sure that taxes on middle-class families don't go up," Obama said. "And since we all theoretically agree on that, we should get that done. If we get that done, a lot of the other stuff is going to be a lot easier."
Obama wants to let tax rates for income over $250,000 for families or $200,000 for individuals return to higher 1990s levels, while maintaining current rates for the rest of the country.
While Cole and other Republicans now say to let that happen, Boehner made clear Wednesday he wanted to keep his strongest bargaining chip.
"I told Tom earlier in our conference meeting that I disagreed with him," Boehner said of Cole, who he described as a "wonderful friend" and strong supporter. "The goal here is to grow the economy and control spending. You're not going to grow the economy if you raise taxes."
At the same time, Boehner reiterated his willingness to include increased tax revenue, as long as it doesn't come from higher rates and is part of a deal that includes spending cuts and entitlement reforms.
Later Wednesday, Obama met with chief executives of major corporations.
Congressional Republicans and Democrats talked separately with deficit-reduction gurus, including former White House Chief of Staff Erskine Bowles, who co-chaired a special panel appointed by Obama in 2010 to study the matter.
"We are going to have to have real compromise. Nobody can be an absolutist, nobody can win everything," Bowles told reporters after his meetings.
Meetings and events on both sides showed the high-profile tactics being used to demonstrate to the nation, including financial markets, that a deal can happen
Stocks opened lower Wednesday as investors grew increasingly concerned about whether Congress can adequately and swiftly address the situation. By the end of the day, after reports of Cole's remarks and other developments, stocks recovered to post a solid gain.
With the U.S. economy showing more signs of improvement in its long recovery from recession, economists point to fears about higher taxes in 2013 as a potential threat to rising consumer confidence.
The fiscal cliff resulted from a failure to reach a deficit reduction agreement in the past two years due to longstanding differences between Democrats and Republicans on taxes -- particularly whether to extend tax cuts from President George W. Bush's administration.
Obama made the issue a central theme of his election campaign, and now the White House believes the president's re-election validated his call for including more tax revenue in addressing the nation's chronic federal deficits and debt.
Last week, Obama's former campaign manager, Jim Messina, said the president's re-election campaign and its grass-roots resources will "live on," most likely as a tool to promote the president's second-term policies.
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 November 16.
Instead, Obama met Tuesday with small business owners to launch his week of campaign-style events.
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.
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.
A 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%.