And Senator Mitch McConnell of Kentucky, the Republican leader and procedural maestro, was pushing his plan that would allow a debt limit increase to clear Congress without Republican fingerprints — and without the guaranteed cuts many in his party are demanding. He would establish an elaborate process where Congress would vote to disapprove instead of approve a debt limit request. That would allow the president to raise the debt ceiling via a successful veto of the disapproval if it came to that.

Despite resistance from conservatives and the initial unease many lawmakers expressed at such a slippery approach, the McConnell gambit was gaining credence as the best escape hatch. Senate Democrats went virtually silent on the idea for fear of jinxing it. While the White House said it was not the preferable option, it was viewed inside the West Wing as a real option nonetheless, even if it would transfer to Mr. Obama and his party all the political responsibility for a debt limit increase.

Some of Mr. McConnell’s colleagues were coming around to it as the reality of a possible default began to sink in.

“I strongly support Senator McConnell’s efforts to avoid a default on our nation’s debt, and the last-case emergency proposal he outlined yesterday to ensure that Republicans aren’t unduly blamed for failure to raise the debt ceiling,” said Senator John McCain, Republican of Arizona.

But with House Republicans showing little to no appetite for Mr. McConnell’s plan, top lawmakers in both parties were looking for ways to sweeten the deal, perhaps by adding required spending cuts or somehow forcing consideration of a deficit-reduction package. Mr. McConnell portrayed his proposal as a last-stand way to spare Republicans from being blamed for a default if no alternative plan could be approved.

Recounting how the 1995 government shutdown helped President Bill Clinton win re-election the following year, Mr. McConnell said any impasse that drove down the nation’s credit rating and led to government checks being delayed could have the same result for Mr. Obama.

“He will say Republicans are making the economy worse,” Mr. McConnell said in an interview with the conservative radio host Laura Ingraham. “It is an argument that he could have a good chance of winning, and all of the sudden we have co-ownership of the economy. That is a very bad position going into the election.”