“I like John Boehner,” Mr. Reid added. “I do feel sorry for him.”

Earlier this week, Representative Eric Cantor of Virginia, the No. 2 House Republican, proposed a two-step resolution to the fiscal impasse that was temporarily pushed into the background by Mr. Obama’s request for approval to initiate a military strike on Syria, since delayed.

Under Mr. Cantor’s plan, the House would have voted this week on a stopgap spending bill to keep the government operating through mid-December at the current level, which reflects the sharp across-the-board cuts known as sequestration. That bill would have a companion resolution to withhold all money for the health care law, but the Senate could simply ignore that resolution and approve the short-term spending bill.

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.