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Some Republicans have backed that legislation, saying they didn’t want to be seen holding tax cuts for the middle class hostage.

But the bigger problem for Mr. Boehner and his chief lieutenants is trying to keep his right flank from fleeing.

“This may surprise you, but we were not consulted on the proposal,” Rep. Louie Gohmert said Tuesday. He said the framework reminded him of the outline Mr. Boehner and Mr. Obama had been working on during last summer’s debate on raising the debt ceiling.

“That’s not a good place to start, I wouldn’t think,” the Texas Republican said, adding that he understands what Mr. Boehner is trying to do, but that every time he extends a hand to the White House, it gets slapped back.

“So why not fall back on principle?” Mr. Gohmert said.

Rep. Steve King, Iowa Republican, said Mr. Boehner’s proposal is better than Mr. Obama’s offer last week, which called for $1.6 trillion in tax increases coupled with $400 billion in potential cuts to health care entitlement programs and hundreds of billions of dollars more in stimulus spending.

But Mr. King said the Republican leadership plan doesn’t strike the right balance on policy either.

“In the end, the speaker’s got to be looking at, ‘Where are 218 votes going to come from?’ And the closer he gets to the president’s proposal, the harder that is to get [it] out of the Republican conference,” Mr. King said. “That’s just a fact.”

Mr. Boehner, though, has said he expects to be able to deliver the votes if and when he strikes a deal with Mr. Obama.

Some House Republicans, while critical of the plan to increase taxes, said they were willing to see where the speaker’s negotiations lead.

“Right now, he’s kind of in the middle of the theater and the rest of us aren’t, and so we have to give him some flexibility in terms of discussion,” said Rep. Jack Kingston, Georgia Republican. “But I’m a guy who believes it’s a spending problem, not a revenue problem.”

Rep. Michael C. Burgess of Texas likewise said the party has to play the cards it is dealt and the leadership is doing the best it can — but that he wasn’t about to compromise his principles.

“Look, the president talks about a mandate. I received a mandate also,” he said. “I did not run on a platform of raising taxes on anyone — top 2 percent, bottom 2 percent — I did not run on that platform. My obligation is to the mandate that I received, and I don’t consider that digging my heels in. I consider that representing my constituents.”