Stephen Crowley/The New York Times

The Senate Republican leader, Mitch McConnell of Kentucky, said Tuesday that a bipartisan budget deal with President Obama was probably out of reach, and he proposed a plan under which the president could increase the federal debt limit without Congressional approval for offsetting spending cuts.

Mr. McConnell’s proposal reflected a growing sense of pessimism on Capitol Hill about the prospects that Mr. Obama and Congressional leaders could come to terms on a budget deal before the government’s borrowing authority hits its limit on Aug. 2. The negotiators sat down for another round of talks at the White House on Tuesday afternoon.

In an interview with CBS News, Mr. Obama said he “cannot guarantee” that the government can pay benefits next month to Social Security recipients, veterans and the disabled if Congress does not increase the federal debt limit, raising the political stakes even as Republicans hardened their opposition to him.

Mr. McConnell’s proposal would give Mr. Obama sweeping power to increase the government’s borrowing authority, in increments, by up to $2.4 trillion — enough, it is estimated, to cover federal obligations through next year — only if Mr. Obama specified spending cuts of equal amounts. But Congress would not have to approve the spending cuts prior to the debt-limit increase.

It is not clear whether House Republicans would sign on to such a measure, given their drive to extract deep spending cuts in return for any debt-limit increase.

Mr. McConnell, who after the midterm elections in November said Republicans’ goal would be to make Mr. Obama a one-term president, said in his Senate speech, “After years of discussions and months of negotiations, I have little question that as long as this president is in the Oval Office, a real solution is probably unattainable.”

At the White House, the press secretary, Jay Carney, responded, “This president’s going to be in office for at least another 18 months, and I think the American people expect Congress to work with him.”

With the Aug. 2 deadline for raising the government’s $14.3 trillion debt limit just three weeks away, Tuesday seemed to be a new low in the summer’s maneuvering between the White House and Congressional Republicans over a debt-reduction package that would clear the way for a vote on the debt limit. The back-and-forth had people in both parties nervously eyeing the financial markets, fretful that the political fighting would unnerve investors; to date, Wall Street has been generally complacent, believing that the White House and Congress ultimately will come to an agreement, if only because failing to do so would hold such economic peril.

Representatives of major business groups, including the U.S. Chamber of Commerce and the Business Roundtable, put out public statements on Tuesday urging a bipartisan agreement on raising the debt limit. They had met with the Treasury secretary, Timothy F. Geithner, on Monday, and other business groups and individuals similarly have contacted the White House to volunteer help in reaching a compromise.

Mayor Michael R. Bloomberg opened an unrelated event on Tuesday morning in New York by saying: “If America, for the first time in its history, defaults on its obligations, it would have a catastrophic effect on our financial system and on our credibility around the world. It would also take a serious toll on our economy, and that at a time when the nation is still trying to recover from the deep recession.”

“America’s good name and credit are just too important to be held hostage to Washington gridlock, and I hope that in the end cooler heads will prevail and an agreement will be reached quickly,” Mr. Bloomberg added.

In advance of what have become daily White House meetings, Congressional Republican leaders stood firm against raising taxes on the wealthy and businesses after 2013, as Mr. Obama demands as part of a “balanced package” with deep spending cuts, including in Medicare and Medicaid.

Mr. Obama, responding to a question in an interview with CBS News, said: “I cannot guarantee that those checks go out on Aug. 3 if we haven’t resolved this issue. Because there may simply not be the money in the coffers to do it.”

He added, “This is not just a matter of Social Security checks. These are veterans’ checks, these are folks on disability and their checks. There are about 70 million checks that go out.”