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Republicans have demonstrated “there is a middle-ground solution that can cut spending and bring in revenue without hurting American small businesses,” Mr. Boehner said in a statement. “It’s a solution President Obama himself once supported. If the math worked in 2011, why doesn’t it work today?”

Mr. Obama didn’t lay out the specifics of his 2011 plan at the time — exactly which deductions and loopholes he would target to produce the revenue. He is now criticizing Republicans for failing to lay out the details of their plan.

But in making his offer to Mr. Obama this week, Mr. Boehner cited a nine-page report by the Committee for a Responsible Federal Budget, a nonpartisan, business-backed group that provided three options for finding the necessary revenue by limiting deductions without increasing tax rates.

White House officials said Republicans are not telling the full story about Mr. Obama’s 2011 quote. The president’s comment, they said, was part of an agreement with Republicans to spend the next year overhauling the tax code to find additional revenue, and if Washington couldn’t get all the complex tax changes through Congress by the end of 2012, tax rates on the top 2 percent of Americans would go up.

Jason Furman, deputy director of the National Economic Council and an assistant to the president for economic policy, also said the White House has “done a lot of work” in the past year in crunching numbers and now “knows more about tax reform” than it did in 2011.

“We didn’t sit here twiddling our thumbs,” Mr. Furman told reporters during a briefing Wednesday.

“So just at a purely analytical level, we know more about tax reform, we know more about tax expenditures, we know more about all of those topics now than, we’ve, you know, known before,” he said.

One of the items that the president’s economic team evaluated was the idea of capping deductions at $25,000 for all Americans. But that proposal, Mr. Furman said the White House found, would raise taxes by an average of $2,400 on 17 million middle-class households and would create a serious disincentive for all Americans who hit that threshold to contribute to charities because they wouldn’t get a tax benefit for doing so.

Placing the limit on households making more than $250,000 would generate only $800 billion, Mr. Furman said, and would create an immense “cliff,” in which someone whose income rose to $251,000 could suddenly owe thousands of dollars more in taxes.