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Mr. McConnell’s proposal is a complicated legislative maneuver that would, in effect, let the president increase the nation’s debt limit with only Democratic votes and subsequently allow Republicans to wash their hands of the politically poisonous issue.

Under the scenario laid out by Mr. McConnell, Mr. Obama would request an increase in the debt ceiling of up to $2.4 trillion in three separate submissions by next summer. The divided Congress likely would reject those requests, which the president then presumably would veto. If Congress failed to muster the two-thirds vote needed to override a veto, the debt ceiling would effectively be lifted by the amount Mr. Obama requested.

The plan would require Mr. Obama to submit spending cuts along with his borrowing requests. But unlike the increase in the debt limit, they wouldn’t automatically take effect.

Mr. McConnell called his plan a “sort of last-choice option.”

“This is not my first choice,” he said. “My first choice is to get an agreement with the president to significantly reduce spending. And we’re going to continue to talk to them about that in the hopes that we can get there.”

Some conservative activists have accused Mr. McConnell of surrendering to Democrats and the president.

“If Mitch McConnell thinks caving to President Obama and allowing him to raise the debt ceiling without cuts is the way to become Senate majority leader he is sorely mistaken,” said conservative activist Brent Bozell. “The American people elected him to serve as a check on Obama’s appetite for out-of-control spending, not to write him a blank check to continue the binge.”

Even as Republicans try to force political pain for a debt limit increase onto Mr. Obama, the White House pushed back.

“The president doesn’t have a vote in this,” said White House press secretary Jay Carney. “It’s Congress that has to act.”

Mr. Carney said if the debt limit is breached, the decisions the government would have to make about payments would amount to “a Sophie’s choice situation.”

Mr. Carney said questions about the kinds of cuts possible if the debt limit is breached should be posed to the Treasury Department, but that the president was saying all programs are potentially at risk.

“We can’t guarantee, if there were a default, any specific bill will be paid,” he said, cautioning that the White House still thinks a deal will be reached.