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Treasury Secretary Timothy F. Geithner in an interview on CNBC on Wednesday echoed the president’s words.

“We are not prepared to have the American economy held hostage to periodic threats that Republicans will force the country to default on our obligations,” he said.

Instead, the White House wants to make permanent a system used during last year’s $2.2 trillion debt-limit hike. Designed by Mr. McConnell, the mechanism requires the president to submit a formal notice to Congress of the need to lift the debt ceiling.

Congress can allow the borrowing to increase by doing nothing, or it could pass a resolution of disapproval, which the president has the power to veto.

But Mr. McConnell said Republicans would fight any attempt to cement last year’s one-time method into permanent law, calling it a “power grab that has no support here.”

Last year, Republicans conceded to the one-time debt-limit mechanism only because the White House signed off on spending cuts that nearly equated the increase in the debt ceiling, but Mr. Obama, running on the momentum from his re-election, says he’s not willing to agree to such a scenario this time.

With both sides digging in, Sen. Richard J. Durbin of Illinois, the second-ranking Democrat in the chamber, said he wouldn’t dismiss the constitutional option as quickly as the White House had.

“I don’t think they ought to rule it out,” Mr. Durbin said. “After the Congress has voted to incur the debt, after the votes are on the record, to ask to borrow the money to pay that debt is obviously a reasonable conclusion.”