A growing chorus of Republicans is urging House leaders to abandon their staunch opposition to higher tax rates for the wealthy with the aim of clearing the way for a broad deal that would also rein in the cost of federal health and retirement programs.

With less than a month before the “fiscal cliff” deadline, President Obama remains adamant about allowing tax rates to rise for the wealthiest 2 percent of taxpayers. Without such a deal, he is “absolutely” ready to go over the cliff, Treasury Secretary Timothy F. Geithner said Wednesday on CNBC.

Many GOP centrists and some conservatives are calling on House Speaker John A. Boehner (R-Ohio) to concede on rates now, while he still has some leverage to demand something in return. Republicans are eager to win changes to fast-growing safety-net programs, such as raising the eligibility age for Medicare and applying a less-generous measure of inflation to Social Security benefits.

After Dec. 31, tax rates for most Americans, including the wealthy, are set to automatically rise, and this could cost Republicans a key bargaining chip in winning changes to entitlements.

“I and some others are advocating giving the president what he wants,” said Rep. Steven C. LaTourette (R-Ohio). But he stressed that this must be part of a package that slows federal borrowing and reduces the debt by $4 trillion to $5 trillion.

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“Quite frankly, some people in this 2 percent who call me, they’re more worried about the fiscal cliff than about the rates going up a couple points. That has bigger risk for them,” said LaTourette, a close Boehner ally who is retiring in January.

Rep. Thomas J. Rooney (R-Fla.) added: “If there are truly real entitlement reforms that are going to preserve Social Security and Medicare for generations to come, it’s going to be very difficult for me to oppose” higher rates for the rich.

An agreement to raise the top tax rate above the current 35 percent would mark a major concession for a Republican Party that has made opposition to higher tax rates a touchstone for more than two decades.

The step would come on top of what was already a significant compromise for the GOP: an offer earlier this week to increase tax revenue by $800 billion over the next decade. That offer involved generating new revenue by closing loopholes and ending deductions for top earners, not by increasing rates.

Republicans are growing alarmed that they could be blamed if Washington is unable to resolve the fiscal stalemate and $500 billion in year-end spending cuts and tax increases kick in. Nearly 90 percent of U.S. households would face higher taxes, and economists warn that the economy could be jolted back into recession.

With talks between the parties apparently at a standstill, Obama called Boehner on Wednesday, their first conversation in a week, aides to both men said.

Meanwhile, Republican leaders were looking for a way out of their political bind.

One option under discussion would entail setting the top tax rate above 35 percent but below the 39.6 percent level that was in effect during the Clinton administration. That compromise could let both parties claim victory. But House Ways and Means Committee Chairman Dave Camp (R-Mich.) said he was reluctant to draft such a plan unless the White House agreed to a tax-revenue target well below the $1.6 trillion Obama has demanded over the next decade.

“Despite dancing in the end zone, which he’s doing, he keeps moving the goal posts. His revenue number keeps changing,” Camp said. “There is a point that the economy can only sustain so much revenue being taken out of it.”

Another possibility would be to give in now on tax rates for the wealthy and tackle Social Security and Medicare early next year, when Obama will need Congress to raise the $16.4 trillion limit on federal borrowing. That could blunt the worst economic effects of the fiscal cliff and push off the fight over spending to a moment when many Republicans think they would have additional leverage.

But there are problems with that strategy, too. Many Republicans say they are willing to vote for higher tax rates only as part of a big debt-reduction deal. So, in the absence of such a deal, top GOP aides have been looking for a way to make Democrats bear the political burden of raising taxes on the rich.

It could work like this: The Republican-controlled House could adopt two competing bills. One, supported primarily by Republicans, would extend the expiring low tax rates for all households, including the rich. The second, supported primarily by Democrats, would extend the current low rates only on income less than $250,000 a year, allowing rates for the wealthy to increase. Both measures would go to the Democratic-controlled Senate, which would then pass only the Democratic bill.

On Thursday, Senate Minority Leader Mitch McConnell (R-Ky.) proposed that the Senate hold an immediate vote on a proposal from Obama that would essentially give the president the ability to unilaterally raise the nation’s debt ceiling.

“Look: the only way we ever cut spending around here is by using the debate over the debt limit to do it,” McConnell said on the floor of the Senate. “Now the president wants to remove that spur to cut altogether.... I assure you: It’s not going to happen. The American people want Washington to get spending under control. And the debt limit is the best tool we have to make the president take that demand seriously.”

The maneuver was designed to force Democrats to go on record as agreeing to largely removing Congress’ role in the sensitive issue of the nation’s legal borrowing limit. Republicans believe many Democrats facing tough reelection battles in 2014 would not vote to hand that kind of power to the president. The measure is not likely to draw the Republican support to muster the 60 votes necessary to pass the Senate, meaning Democrats would be forced to take a tough vote on a bill doomed to failure.

Obama had proposed the mechanism as part of a deficit reduction package submitted to Congressional leaders by Geithner last week. Obama suggested that in the future, the president would propose increases in the debt limit in a resolution that Congress could then vote to disapprove. The president could then veto the disapproval resolution — allowing the debt limit to rise unless Congress could muster the two-thirds vote necessary to override his action.

Senate Majority Leader Harry M. Reid (D-Nev.) said he would examine McConnell’s proposal, and later in the day Reid called for a vote, but on certain terms: It would be an up-or-down vote, allowing the Senate to skip to a final vote so that the measure could be approved on a simple majority.

Concerned that the bill could be approved, McConnell blocked the vote.

On Wednesday, Obama warned Republicans not to pursue a course that would set up a fresh battle over the debt limit. Addressing business leaders, he recalled “the catastrophe that happened in August of 2011,” when Republicans last used the debt limit as a point of leverage. The nation came within days of default and consumer confidence plummeted.

“That is a bad strategy for America, it’s a bad strategy for your businesses, and it is not a game that I will play,” Obama said in an address to the Business Roundtable.

He argued that “we can probably solve this in about a week” if Republicans give ground on tax rates. For the second day in a row, he raised the prospect of a compromise that sets the top rate lower than 39.6 percent.

“We’ve seen some movement over the last several days among some Republicans,” Obama said. “I think there’s a recognition that maybe they can accept some rate increases as long as it’s combined with serious entitlement reform and additional spending cuts.”

In recent weeks, influential conservative pundits have begun making similar arguments. The Wall Street Journal stated in an editorial last month that Republicans could score political points by keeping the top rate below 39.6 percent.

Such views are reverberating on Capitol Hill, where Sen. Tom Coburn (R-Okla.), a champion of smaller government, on Wednesday became the latest voice to join the chorus.

“Personally, I know we have to raise revenue. I don’t really care which way we do it. Actually, I would rather see the rates go up than do it the other way, because it gives us a greater chance to reform the tax code and broaden the base in the future,” Coburn said on MSNBC.

Rep. Jeb Hensarling (R-Tex.), a conservative veteran of recent budget wars, also acknowledged that tax increases are coming.

“The president is going to get some form or fashion of revenue. That’s baked into current law,” he said. “I didn’t vote for it. I don’t want it. But I recognize it, and House Republicans will work to minimize the damage to the economy.”