Under a "discharge petition," a bill can be brought to the floor without going through a committee or without approval of House leadership. The bill would need majority support -- or 218 votes -- to pass.
In the current lame-duck session of Congress, there are 192 Democrats in the House, so at least 26 Republicans would have to defect for Pelosi's motion to succeed.
Cole and fellow conservative Rep. Tim Scott both said last week they believed that Senate tax measure would pass if brought to a vote in the House.
However, Cole made clear he would follow the directions of the leadership, tamping down the chances of a GOP revolt against Boehner over the issue.
Boehner has so far rejected any increase in tax rates, even if only for the wealthiest Americans, saying his offer to include additional revenue by eliminating some loopholes and deductions was a major concession.
On Sunday, he described the latest Democratic proposal for a broader deficit reduction plan as one-sided and unworthy of discussion.
"The president's idea of a negotiation is, 'Roll over and do what I ask,'" Boehner told Fox News Sunday, describing himself as "flabbergasted" by the plan put forward last week by Treasury Secretary Timothy Geithner. "I looked at him and I said, 'You can't be serious.'"
Such posturing on both sides reflects the mistrust built up over two years of deficit wars that have left Congress with a reputation of dysfunction.
Failure to reach a deal to avert the fiscal cliff and devise a framework for a broader deficit reduction package to be negotiated when the new Congress is seated in January would cause certain economic turmoil and threaten the U.S. credit rating.
The Bush tax cuts of 2001 and 2003 already were extended by two years as part of budget negotiations earlier in Obama's term.
In addition, spending cuts of $1 trillion over 10 years also kick in next year, the result of the Republican-led standoff over raising the federal debt ceiling in 2011 that led to a U.S. credit rating downgrade.
Sharp tax increases and automatic spending cuts would cut the projected federal budget deficit nearly in half -- but also would threaten millions of jobs, especially those dependent on government contracting, and risk a return to recession, according to the non-partisan Congressional Budget Office.
The non-partisan Tax Policy Center estimates middle class families would pay about $2,000 a year more in taxes without another extension of all the Bush tax cuts, as sought by Republicans.
Without a deal in the lame-duck Congress, each side wants the other to get the lion's share of public blame for higher taxes and economic uncertainty.
The plan that Geithner presented to Republicans last week called for $1.6 trillion in new taxes, including the rate hike for income over $250,000.
"There's not going to be an agreement without rates going up," Geithner told CNN in an interview aired on Sunday. "If they are going to force higher rates on virtually all Americans because they're unwilling to let tax rates go up on 2 percent of Americans, then, I mean that's the choice they're going to have to make.
Obama also wants to close loopholes, limit deductions, raise the estate tax rate to 2009 levels and increase taxes on capital gains and dividend taxes.
In addition, the Obama proposal calls for a new $50 billion stimulus package, a home mortgage refinancing plan, and an extension of unemployment insurance benefits. It also would extend the payroll tax cut passed early in Obama's administration.
In return, multiple sources told CNN that Obama is offering $400 billion in new cuts to Medicare and other entitlement programs -- with specifics decided next year.
According to a CNN/ORC International Poll released last week, 56% of people said taxes should be kept high to help lower-income people, while 36% said taxes should be kept low to create jobs.
Another survey, by ABC News and the Washington Post, showed two-thirds of respondents supported Obama's call for holding down tax rates for everyone but the wealthiest Americans.
Mark Zandi, chief economist for Moody's Analytics, predicted on the CBS program "Face the Nation" that lawmakers would reach a short-term deal that would limit the economic damage, extend the U.S. debt ceiling to avoid another standoff like the one in 2011 and lay down a framework for future deficit reduction talks.
CNN's Ashley Killough, Adam Levy and Deirdre Walsh contributed to this report.
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