Then the House would vote to raise the debt ceiling enough for a year of borrowing, but demand a year's delay in carrying out the health care law.
Within 24 hours, the House's most ardent conservatives revolted, declaring the defunding resolution a gimmick that fell well short of their drive to undo the health care law. House Democrats said they would oppose not only stripping the health care law of money but also a spending level that maintains sequestration.
"The continued operation of the sequester is inimical to the interest of the United States, to the government, to the people and to international security," said Representative Steny H. Hoyer of Maryland, the Democratic whip, who promised to hold his members against the Cantor plan.
It was delayed indefinitely as House Republicans resumed their search for a measure that could unite them. One group of conservatives on Thursday pressed what they called a compromise: a one-year stopgap spending bill that would raise the debt ceiling for a year, delay all aspects of the health care law for a year, and give back some of the Pentagon cuts as a sweetener.
Backers insisted on Thursday that it was a package Mr. Obama should be able to accept. Representative Phil Gingrey, Republican of Georgia, said seven Democratic senators facing re-election fights next year in Republican-leaning states would provide a beachhead of Democratic support, and noted the president had already agreed to some delays for his health law.
(Read more: US Treasury to hit debt limit by mid-October)
Democrats scoffed at the Republican plans, and even some Republican leadership aides questioned how any could get to the president's desk. Mr. Reid called the succession of proposals "juvenile political games" and suggested that many Republicans had lost touch with reality.
Senator Richard J. Durbin of Illinois, the second ranking Democrat, said, "Sometimes I sympathize with Speaker Boehner, but the fact of the matter is, if he wants to lead for the good of the nation, he has to step beyond the Tea Party faction of his caucus."
Republican divisions were manifest not only in the tactics they have proposed but also in the strategic aims of those tactics. Mr. Boehner continued to emphasize taming the budget deficit as the price for a debt-ceiling increase.
But the urgency of that mission was undercut by government financing figures released Thursday by the Treasury, which showed the smallest annual shortfall since 2008. In the first 11 months of the current fiscal year, the deficit reached $755.8 billion, with tax revenues rising and spending falling. The deficit in fiscal 2012 was $1.1 trillion.
With no resolution in sight, Republican leaders said decisions would have to be made next week on a way forward—with Democratic votes, or Republican unity. But Mr. Boehner gave no indication he knew which way to turn.
"There are a million options that are being discussed by a lot of people," he said. "When we have something to report, we'll let you know."
—By Jonathan Weisman of The New York Times