WASHINGTON — With the clock ticking down, President Obama and Congressional leaders began a final effort to forge a broad deficit-reduction plan even as new cracks appeared among House Republicans over how to proceed.

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.