The White House suggested for the first time that Mr. Obama might be willing to agree to a short-term increase in the debt ceiling by Congress — perhaps for a few days — if such a deal was in sight, stepping up the pressure on the two parties to come to terms.

Mr. Obama met separately at the White House with Republican and Democratic leaders. But neither side reported any substantive progress as they searched for a formula that would include deep spending cuts, cost-saving changes to entitlement programs and an overhaul of the tax code that would increase revenues by closing certain tax breaks and eliminating deductions but also lower some tax rates.

Politically, the main question remained whether House Republicans would be willing to negotiate over any package that could be construed as raising taxes, and throughout the day there were signs of internal debate among party leaders.

Speaker John A. Boehner has shown continued interest in a deal if it can be done in a way that emphasizes lower tax rates.

But Representative Eric Cantor, the No. 2 Republican, and others like Representative Paul D. Ryan of Wisconsin, the Budget Committee chairman, warned that the most specific proposal to be made public so far — and the one that has done the most to reopen the possibility of a bipartisan accord — relied far too much for them on higher revenues to cut projected deficits.

That plan is the one put forward Tuesday by the so-called Gang of Six, a bipartisan group of senators who worked for months to reach an agreement and whose work was lauded by Mr. Obama as a sign that a deal was possible. The plan included a net increase in government revenue of about $1 trillion over a decade.

“I am concerned with the Gang of Six’s revenue target,” Mr. Cantor said.

Deepening the impasse was growing opposition among House Republicans to a fallback position developed by Senator Mitch McConnell, the Republican minority leader, that would allow the debt ceiling to be raised without any support from Republicans but also would not impose the dollar-for-dollar cuts in spending that have been central to the party’s negotiating position.

But given the looming prospect of Congress’s missing the Aug. 2 deadline and risking a default on United States government debt — a development that could shake financial markets and harm the already weak economy — some Republicans appeared more willing to consider a deal locking in spending cuts that Mr. Obama has said he would take if balanced by new revenues. Conservatives have split over sticking to their no-taxes principles if it means walking away from progress toward restraining the growth of government.

At the White House, officials alternated between a sense of encouragement and growing concern that time is running out to avert a full-scale crisis.

“There is still time to do something significant if all parties are willing to compromise, because the parameters of what that might look like are well known,” said Jay Carney, the White House spokesman.

With an eye on the calendar, the president summoned leaders of both parties to build on Tuesday’s release of the $3.7 trillion deficit-cutting plan by the Senate’s Gang of Six.

At the Capitol, however, the emphasis was on a plan that Mr. McConnell has been putting together with the Senate majority leader, Harry Reid, Democrat of Nevada, to empower Mr. Obama to raise the borrowing ceiling.

The four House leaders — Mr. Boehner, Mr. Cantor; Representative Nancy Pelosi of California, the Democratic leader; and Representative Steny H. Hoyer of Maryland, the No. 2 Democrat — met Wednesday and, according to officials, reviewed problems with the McConnell plan.

Such talks between the two parties’ leaders in the House, which are rare given the polarized relations in that chamber, reflect the recognition that the Republican majority cannot pass the increase in the debt limit without a significant number of votes from Democrats.

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