WASHINGTON — With his bill to repeal the Affordable Care Act in deep trouble, Mitch McConnell, the Senate majority leader, raised an alternate possibility on Tuesday: Either Republicans come together, or he would have to work with Democrats to shore up the deteriorating health law.

That raised a tantalizing prospect: bipartisanship.

The idea is not so far-fetched. For years, Republicans and Democrats have explored avenues for changing or improving President Barack Obama’s health care law, from tweaks to the requirement for employers to offer health insurance to revisions involving how the marketplaces created under the law operate.

Mr. McConnell had hoped the Senate would pass the repeal bill this week, but he met resistance from moderate and conservative members of his caucus. He spent much of his time on Wednesday in discussions with Republican senators, seeking agreement on the substance of a revised bill.

Republican senators said that Mr. McConnell wanted to finish work on the legislation by Friday, submit it for analysis by the Congressional Budget Office and then press the Senate to take it up after a weeklong break for the Fourth of July.