Conservative Rep. Tim Scott of South Carolina also said he thinks the Obama tax plan would pass the House, though he made clear to CNN he would oppose it.
Obama made clear Wednesday that he hopes public pressure will cause House Republicans to move from their unyielding stance.
"The lesson is that when enough people get involved, we have a pretty good track record of making Congress work," he said.
Cole and some other conservatives say such pressure is the reason to simply give the president what he wants and move past the immediate tax issue.
"If we agree that taxes shouldn't go up on 98% of the people, shouldn't we take that now and get that set aside and make sure they know their taxes aren't going up?" Cole said Wednesday night on CNN's "Anderson Cooper 360."
Sounding a lot like Obama, Cole said that "if we can give assurance to most Americans that their taxes are going to be fine, I think that's helpful to them in planning their lives going forward."
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."
Boehner outlined a similar process on Thursday, but demanded more commitment from Obama and Democrats on spending cuts and entitlement reforms.
His framework includes what he called a "down payment" for the rest of this year that would include spending cuts and additional tax revenue, but not higher tax rates. That would set up negotiations on tax reform and other aspects of deficit reduction next year, Boehner said.
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 impending fiscal cliff resulted from a failure to reach a deficit reduction agreement in the past two years due to long-standing differences between Democrats and Republicans on taxes -- particularly whether to extend tax cuts from President George W. Bush's administration.
Republicans seeking to shrink the size of government oppose allowing any tax rates to return to pre-cut levels, arguing that Obama's plan would hinder job growth because some small business owners who file personal returns would pay higher taxes under it.
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 would include entitlement reforms and spending cuts.
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.
New polls this week, including one by CNN/ORC International, showed a solid majority of respondents supports the Democratic stance that any agreement should include a mix of spending cuts and tax increases. An ABC News/Washington Post survey showed a strong majority favoring the Obama tax proposal to raise tax rates on the wealthy.