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Mr. McConnell added additional fuel to the political fire this week by offering a Tax Hike Prevention Act, which would extend all the Bush-era tax cuts, limit the alternative minimum tax and reform the so-called “death tax.”

But Mr. Hoyer and a growing chorus of Democrats argue that the McConnell proposal would cost a whopping $3.9 trillion over 10 years and that he has come up with only $300 billion to pay for it.

It “will cost more than all of the legislation that the Bush administration asked for to meet the economic crisis or anything that together we have done over the last 20 months of this Congress,” Mr. Hoyer said.

Heading into Wednesday, Mr. Obama and Democratic leaders have suggested there is no wiggle room when it comes to their opposition to extending the cuts for households earning more than $250,000 a year. The president maintained that it would cost $700 billion over 10 years to extend the tax breaks for wealthier Americans, something the country simply can’t currently afford.

But, following Mr. Hoyer’s move on Wednesday, it appears the tone is shifting on Capitol Hill and that there could be growing opposition to Mr. Obama’s plan.

Mr. Hoyer admitted as much on Wednesday after he was asked what he was doing about members who oppose the White House’s plan to end tax cuts for the top 2 percent to 3 percent of income earners. “Every member needs to take their own position on this issue to what they think is appropriate,” he said.

The debate picked up speed on Sunday after Mr. Boehner, Ohio Republican, said he is prepared to support Mr. Obama’s plan to extend tax cuts for just the lower and middle classes, even if it does not include Republican-backed cuts for wealthier Americans.

“If the only option I have is to vote for some of those tax reductions, I’ll vote for it,” Mr. Boehner said on CBS’ “Face the Nation.” “But I’ve been making the point now for months that we need to extend all the current rates for all Americans if we want to get our economy going again, and we want to get jobs in America.”

The remarks followed a campaign-style rally last week in Cleveland, Mr. Boehner’s backyard, where Mr. Obama called to extend the tax cuts for those earning less than $250,000 and attacked the Republican House leader for bringing “no new ideas” to the table for economic policy.

On Wednesday, Mr. Hoyer suggested to reporters that Mr. Boehner made the “mistake of a few short seconds of reasonableness,” which led some, including the Wall Street Journal, to suggest he may be unfit to be the next speaker of the House should Republicans win the chamber in the coming elections.

Mr. Boehner, meanwhile, continued to qualify his comments on Wednesday, saying, “Raising taxes on anyone, especially small businesses, is the exact wrong thing to do in a struggling economy. Economists agree and a growing number of Democrats agree.”