The leader, Senator Mitch McConnell of Kentucky, said a bipartisan budget-cutting deal is probably out of reach, making it unlikely that Republicans would approve an increase in the government?s debt limit by Aug. 2. To prevent default, he proposed that Congress in effect empower Mr. Obama to raise the government?s borrowing limit without its prior approval of offsetting cuts in spending.

Administration officials welcomed the McConnell initiative for at least signaling that both parties? leaders were committed to averting a potential economy-shaking government default; many Democrats in Congress saw it as a way to avoid the sort of deep cuts in Medicare, Medicaid and Social Security that Republicans have sought as the price of their votes for a debt-limit increase.

But many conservatives immediately assailed Mr. McConnell?s proposal as a panicky sell-out, much as they in recent days had attacked the House Republican leader, Speaker John A. Boehner, for privately discussing with Mr. Obama a debt-reduction deal that could raise revenues as well as cut spending ? ultimately forcing Mr. Boehner to retreat.

Mr. Boehner, though, suggested he could be open to the McConnell proposal, highlighting concern among the Republican leadership about how the showdown with the administration would end, the blame the party could suffer if a debt-limit crisis hurt the economy and the challenge of leading their rank and file into a compromise.

Mr. Boehner, in an interview on Fox News, said, ?I think everybody believes there needs to be a backup plan if we are unable to come to an agreement, and frankly I think Mitch has done good work.?

Yet Mr. Boehner also acknowledged the limits of his own leadership given the passions of conservatives on the issue. ?I don?t think such a proposal could pass the House in any way, shape or form,? he said. ?You have a number of members who will never vote to raise the debt ceiling and a large block of members who believe this really is the moment to put our fiscal house in order.?

Mr. McConnell made his proposal after days in which Mr. Obama has been on the offensive in the fight over the budget, calling for more deficit reduction than Republicans are and urging both parties to put ideology aside in favor of compromise.

While Mr. McConnell?s plan would face an array of political and perhaps constitutional issues, it signaled that Republican leaders did not intend to let conservative demands for deep spending cuts provoke a possible financial crisis and saddle the party with a reputation for irresponsible intransigence. And with prospects for a broad budget deal having dimmed, Mr. McConnell?s plan would shift both substantive and political responsibility onto Mr. Obama, forcing him to take almost sole ownership of a debt-limit increase and any consequences from not doing more to address the budget deficit.

The longtime conservative activist Brent Bozell encouraged followers online to call Mr. McConnell?s office, saying he had ?betrayed the trust of the American people.? And Newt Gingrich, a Republican presidential candidate, wrote on Twitter, ?McConnell?s plan is an irresponsible surrender to big government, big deficits and continued overspending.? Yet Mr. Gingrich is no stranger to the risks in a showdown with a president of the other party; in the mid-1990s, as House speaker, he forced a government shutdown in a budget fight with President Bill Clinton that backfired against Congressional Republicans.

The conservatives? reaction against Mr. McConnell, coming after their earlier attack on Mr. Boehner, raised the prospect that the White House has long feared ? that Republican leaders cannot lead a rank-and-file membership that is defiantly opposed to compromise with Mr. Obama.

Among administration officials, that fear offset their relief that the McConnell proposal suggested at least a fallback in the event that the talks break down.

?Senator McConnell?s proposal today reaffirmed what leaders of both parties have stated clearly, that defaulting on America?s past-due bills is not an option,? the White House press secretary, Jay Carney, said in a statement. ?The president continues to believe that our focus must remain on seizing this unique opportunity to come to agreement on significant, balanced deficit reduction.?