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WASHINGTON — House Republican leaders struggled on Tuesday to devise a new proposal to reopen the government and alter parts of the president’s health care law after a plan presented behind closed doors to the Republican rank and file failed to attract enough support immediately to pass.

After more than two hours, Republican leaders walked back from a plan that had emerged this morning. Speaker John A. Boehner told reporters there were “no decisions about what exactly we will do.”

“We’re trying to find a way forward in a bipartisan way that would continue to provide fairness to the American people under Obamacare,” he said, but acknowledged that “there are a lot of opinions” among his fractious troops.

The apparent disarray left Mr. Boehner with a crucial decision as time ticked down toward a possible default on government obligations on Thursday. Does he accept whatever bipartisan plan emerges from the Senate, likely on Tuesday, or does he continue to try to get his troops in line behind a counterproposal that still does not exist?

Representative Kevin McCarthy of California, the House majority whip, said Republican leaders were “very cognizant of the calendar.”

But House Republicans appear intent to extract at least one concession, depriving members of Congress, the president, vice president and White House political appointees of subsidies when they buy health insurance under the health care law’s new exchanges. Representative Eric Cantor of Virginia, the majority leader, said any proposal must reflect what he called “our position on fairness” — “no special treatment under the law.”

Those words have become code for legislative language that denies employer matches to politicians forced into the exchanges by a clause in the original health care law.