Negotiations over a deficit-reduction agreement spiraled downward Tuesday as the White House and congressional leaders dug in even as anxiety mounted that they could wait too long to reach a deal to avoid a government default.

In one sign that top leaders worry they won't reach a deal in time, Senate Minority Leader Mitch McConnell (R., Ky.) unveiled a proposal that would allow President Barack Obama to raise on his own the federal borrowing limit by $2.4 trillion in three installments before the end of 2012, unless two-thirds of Congress votes to block it.

Because Mr. Obama would have to lift the debt ceiling, it could place any political fallout on him for doing so. But Republican conservatives protested that Mr. McConnell's plan would give up the leverage the GOP has to force the White House to approve government spending cuts in return for a debt-ceiling increase.

The Treasury says the $14.29 trillion debt cap must be raised by Aug. 2 or the government will run out of cash to pay all its bills. Mr. Obama warned in an interview with CBS News that he could not guarantee that the government would be able to send checks to recipients of Social Security, military pensions and other government benefits next month without a debt-ceiling increase. "There may simply not be the money in the coffers to do it," he said.

Taken together, the developments of the past few days suggest that the collapse of the effort by Mr. Obama and House Speaker John Boehner (R., Ohio) to strike a bigger and more historic deficit-cutting deal has made the effort to negotiate a smaller agreement harder rather than easier.