WASHINGTON — The Senate’s top Republican on Thursday urged the GOP-led House to pass a short-term renewal of payroll tax cuts and break an impasse that threatens all workers with a Jan. 1 tax increase. Within hours, a House Republican freshman broke ranks and agreed, signaling that fierce pressure from almost every corner of the Republican Party had begun to crack conservative opposition to a short-term fix.

“This isn’t about proving a point,” said Rep. Sean Duffy, R-Wis. “Middle class families deserve a Congress that will rise above the squabbling and ensure their taxes don’t go up right after Christmas.”

He called on his own Republican leaders to take up a two-month extension of the cuts, days after the House rejected such a plan passed by the Senate and held out instead for a one-year renewal

The statement suggested that GOP Leader Mitch McConnell’s plan may have started to break down a deadlock between House Republicans and Democrats controlling the Senate that had split House and Senate Republicans, too. This, as President Barack Obama has underscored that only swift action can head off an increase of 2 percentage points in the Social Security tax paid by employees.

McConnell urged House Republicans to pass a new short-term extension while calling on Senate Majority Leader Harry Reid, D-Nev., to appoint negotiators on the separate House measure that would bring a year-long renewal of the payroll tax and jobless benefits. At least two other Republican senators, Olympia Snowe of Maine and Mike Johanns of Nebraska, issued statements Thursday imploring the House to approve the two-month fix.

In competing news conferences and statements, all sides sought to avoid blame should taxes go up Jan. 1, just as Americans begin paying holiday bills. House Republicans in particular were facing fire from GOP establishment figures incensed that they would risk losing the tax cut issue to Democrats at the dawn of the 2012 presidential and congressional election year.

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McConnell’s move intensifies pressure on House Speaker John Boehner, R-Ohio, to cut his losses and agree to a short-term bill. But Boehner is thus far holding firm.

“This is an issue where an overwhelming number of people of both parties agree,” Obama said in an appearance in which he was flanked by several people who had tweeted the White House about how they would be hurt by higher taxes. “Has this place become so dysfunctional that even when we agree to things we can’t do it?” He added: “Enough is enough.”

In a Thursday morning phone call, Boehner urged Obama to send administration officials to the Capitol to negotiate an agreement on a long-term measure demanded by Republicans. Obama declined the offer.

“The president told Speaker Boehner that he is committed to begin working immediately on a full-year agreement once the House passes the bipartisan Senate compromise that prevents a tax hike on 160 million Americans on January 1,” said a White House statement.

McConnell weighed in, saying the House should pass a short-term extension that ensures no disruption in the expiring payroll tax cuts and Reid should appoint negotiators allowing Congress to “work on a solution for the longer extensions.”

McConnell’s move was welcomed by Democrats but received a tepid reaction from Boehner.

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“We believe, as Senator McConnell suggested, the two chambers should work to reconcile the two bills so that we can provide a full year of payroll tax relief — and do it before year’s end,” said Boehner spokesman Kevin Smith. He declined to comment on McConnell’s suggestion that House Republicans back away from their insistence on a year-long extension — or none.

But Rep. Dave Camp of Michigan, the powerful chairman of the tax-writing Ways and Means Committee and a House negotiator on this issue, suggested that he would be open to a three-month extension.

House Republicans have complained that the Senate’s proposed 60-day extension would be hard for companies to implement and would be too complicated for businesses — which file taxes every three months — to manage.

The Senate passed the two-month measure after Reid and McConnell tried but failed last week to come up with a long-term extension. Reid said that if the House passes a short-term measure he’ll restart talks on a year-long plan.

“We have made good progress towards a year-long extension of all of these programs, but there remain important differences between the two parties,” Reid said. “Once the House passes the Senate’s bipartisan compromise … I will be happy to restart the negotiating process to forge a year-long extension.”

McConnell’s idea would require the House to generate a new bill — which could address the flaws Republicans have complained about — and send the measure to the Senate. It would take unanimous agreement by the Senate to pass the measure Tuesday or Friday.

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In an appearance on CNBC’s “Squawk Box,” Camp seemed open to an idea like McConnell’s.

“The policy that would work best for this country is to do a year,” Camp said. “If we cannot do a year, we should at least do a quarter.”

The conflict arose after the Senate, on a bipartisan vote, passed legislation last week to extend for two months the payroll tax cut and jobless benefits. The House had just days before passed a full-year extension that included a series of conservative policy prescriptions.

McConnell was a driving force behind the Senate measure and had been virtually silent in the political firestorm that has erupted since, knocking tea party House Republicans on their heels.

The Republican establishment was putting special pressure on House Republicans who were refusing to compromise. The 2008 GOP presidential nominee John McCain, former Bush administration Karl Rove and The Wall Street Journal editorial page were among the conservative voices urging Republicans on Capitol Hill to get it together.

The impasse also put Republican presidential contenders in an awkward spot less than two weeks before the Iowa caucuses that kick off the nomination process. Former Massachusetts Gov. Mitt Romney refused to be pinned down on the issue, while former House Speaker Newt Gingrich castigated Congress for “an absurd dereliction of duty.”