Updated: 1:29 p.m.

Senate Minority Leader Mitch McConnell (R-Ky.) today offered a potential resolution to the weeklong stalemate over the expiring payroll tax holiday and unemployment benefits, and Speaker John Boehner (R-Ohio) quickly convened a meeting of key Republicans to consider it, all the while reasserting the House GOP’s position.

McConnell called on the House to pass the Senate’s two-month deal and the Senate to appoint conferees for the longer-term deal but did not specify a time frame to conclude the conference.

McConnell’s offer could give House Republicans a face-saving resolution to the impasse, given his House counterparts have been hammered by House and Senate Democrats, rank-and-file Senate Republicans and even media outlets normally friendly to their cause for refusing to even bring the two-month bill to the floor.

Boehner released a statement quickly after McConnell’s offer went public, which reasserted the House Republican position.

“The House and Senate have two different bills, but the same goal. That is why we believe, as Sen. 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 a spokesman for Boehner, just minutes after McConnell released his statement.

Rep. Tom Price (R-Ga.), who was appointed by Boehner to be a conferee on the payroll tax cut bill, exited the Speaker’s office today saying the House is unlikely to back down from its demand that the Senate negotiate a yearlong deal before the new year begins.

“We continue to believe that the differences need to be worked out in a conference committee,” Price said. He added, “Look, there’s a difference between the House and the Senate. The way that you solve these is to find a conference committee, get together and settle out the differences.” He noted the holidays should not be an obstacle, given that many people “all across this country continue to work” through the winter holidays.

McConnell has largely kept mum since the Senate’s Saturday vote to pass a two-month extension of the payroll tax cut and the subsequent House GOP refusal to take up and pass the Senate-brokered bipartisan compromise.

Instead, Boehner at the urging of his caucus, opted to appoint conferees to reconcile the two bills, with just days left on the 2011 calendar.

Today, McConnell proclaimed that neither position was “mutually exclusive.”

“The House and Senate have both passed bipartisan bills to require the President to quickly make a decision on whether to support thousands of U.S. manufacturing jobs through the Keystone XL pipeline, and to extend unemployment insurance, the temporary payroll tax cut and seniors’ access to medical care. There is no reason why Congress and the President cannot accomplish all of these things before the end of the year,” McConnell said in a statement. “House Republicans sensibly want greater certainty about the duration of these provisions, while Senate Democrats want more time to negotiate the terms. These goals are not mutually exclusive.”

Senate Majority Leader Harry Reid (D-Nev.) said he would gladly appoint conferees, as soon as the House passes the short-term deal.

“Once the House passes the Senate’s bipartisan compromise to hold middle-class families harmless while we work out our differences, I will be happy to restart the negotiating process to forge a yearlong extension,” he said.

At a briefing this morning, House Minority Whip Steny Hoyer hesitated when told the news of McConnell’s offer, alluding to the distance between House and Senate Republicans.

“I don’t know whether Sen. McConnell can speak on behalf of Speaker Boehner,” the Maryland Democrat said. “We’ll see what [Boehner] has to say.” That was minutes before Boehner’s statement came out.

Hoyer did note the offer sounded similar to Reid’s position that the two chambers should begin negotiating the one-year extension immediately after the two-month bill is passed.

Indeed, at a news conference this morning before McConnell released his statement, Boehner and his conferees indicated that they are not at all willing to pass a two-month extension.

“Politics will be politics,” Boehner said of the criticism from some in his own party. “Sometimes it’s hard to do the right thing, but we’re fighting to do the right thing.”

Boehner called President Barack Obama this morning, according to a Boehner aide, and proposed that he send members of his economic policy team to Capitol Hill to negotiate.

“The Speaker explained his concern that flaws in the Senate-passed bill will be unworkable for many small-business job creators,” the Boehner aide said. “He reiterated that if their shared goal is a one-year bill, there is no reason an agreement cannot be reached before year’s end. The president declined the Speaker’s offer.”

At the briefing, House Majority Leader Eric Cantor personally called on the president to come to the Capitol.

“I saw the president out yesterday doing his Christmas shopping. I saw he brought his dog with him. You know, we’re here. He can bring his dog up here,” the Virginia Republican said.

Regardless of how the negotiations proceed, or don’t, House Republicans still face a barrage of criticism from their own ranks, as political figures from Republican strategist Karl Rove to presidential candidate and former Speaker Newt Gingrich (R-Ga.) have called on them to back down.

Jonathan Strong contributed to this report.