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“Where the White House is, is they want to make sure that revenue actually comes in — that it’s real. And their belief is the only way you make it real is to have it in the form of higher rates,” Mr. Bowles said at a breakfast hosted by the Christian Science Monitor. “He wants some portion of it, some major portion of it, in the form of higher rates.”

Some top congressional Democrats have drawn another line in the sand in recent weeks, saying they do not want Social Security and Medicare changes to be on the table.

But Mr. Bowles said the president and his party must tackle entitlement spending. He said he laid out for the president at least $600 billion in cuts to health care programs that could be part of a final deal.

Some cracks began to show in Republican unity this week on Capitol Hill, where Rep. Tom Cole, Oklahoma Republican, told colleagues they should accept Mr. Obama’s demand to extend most tax cuts now, and fight over the top tax bracket later.

Mr. Boehner shot down that suggestion.

“The goal here is to grow the economy and control spending. You’re not going to grow the economy if you raise tax rates on the top — on the top two rates,” Mr. Boehner said. “It’ll hurt small businesses. It’ll hurt our economy. It’s why it’s not the right approach.”

Off Capitol Hill, Grover Norquist, head of Americans for Tax Reform and keeper of an influential no-new-taxes pledge, said he doesn’t see signs that the Republicans are ready to abandon the pledge in order to strike a deal with Democrats.

Speaking at a Politico forum, Mr. Norquist said Republicans should look to push the fiscal cliff debate into next year, and then use short-term spending resolutions and incremental bumps in the nation’s borrowing limit to force Mr. Obama to accept spending cuts that he otherwise wouldn’t swallow.

“They can have him on a rather short leash, on a small — you know, ‘Here’s your allowance, come back next month if you’ve behaved,’” he said, adding, “Month if he’s good, weekly if he’s not.”

Over the past weeks, several prominent Republican lawmakers have said they don’t feel bound to their tax pledges anymore, but Mr. Norquist said he was not concerned.

He said those lawmakers have all said they would accept higher taxes only when coupled with the kinds of spending cuts “that Democrats will never give.”