Mr. McConnell faces an increasingly onerous math problem. He can afford to lose only two Republicans if he is to get a bill through the Senate, and that would require the help of Vice President Mike Pence, who would have to cast the tiebreaking vote. But at least three senators in the party are diametrically opposed to the views of at least another three, so the path to agreement is narrow.

Republicans are roughly split over whether the expansion of Medicaid under the Affordable Care Act should be rolled back or continued, at least in the short run. They disagree about how the federal government should grant states more control over setting insurance standards. They are also divided over a critical portion of the House bill, which would allow states to obtain waivers from two of the most important federal mandates: a requirement to provide a minimum set of health benefits and a prohibition against charging higher prices to people with pre-existing medical conditions.

The challenges facing Senate Republicans are so great that overhauling the tax code is starting to look easier by comparison. “I allow that’s a possibility,” said Senator Pat Toomey, Republican of Pennsylvania, who is closely involved in negotiating both issues and favors a rollback of the Medicaid program.