Under the revised bill, it appears that a state could allow insurers to set higher premiums based on a person’s health status, though not on the basis of sex or genetic information. In applying for federal grants, state officials would have to describe how they would “maintain access to adequate and affordable health insurance coverage for individuals with pre-existing conditions.”

All of this might be too late. Changes to the bill’s funding formula to win over Ms. Murkowski might cost it support with conservatives. And the hasty revisions could only strengthen concern that Congress would be moving forward on a bill that has not been properly vetted. The Congressional Budget Office is set to release a partial analysis of its impact as soon as Monday, but that report may not include the latest changes.

Speaking to reporters in New Jersey on Sunday, Mr. Trump seemed to be looking ahead to the next big legislative goal for Republicans — overhauling the tax code — even as he talked up the Graham-Cassidy bill and applied pressure to resistant senators.

“Eventually, we will win on that,” he said of repealing the health law. “My primary focus, I must tell you — and has been from the beginning, as you can imagine — is taxes.”

But the architects of the latest repeal plan, Mr. Graham and Mr. Cassidy, were not giving up.

“We’re moving forward, and we’ll see what happens next week,” Mr. Graham said on ABC’s “This Week.” “I’m very excited about it. We finally found an alternative to Obamacare that makes sense.”

Marc Short, the White House director of legislative affairs, said on NBC’s “Meet the Press” that a Senate vote was planned for this week. But a spokesman for Mr. McConnell declined to affirm that timeline on Sunday.

The Graham-Cassidy bill is the latest attempt to develop a repeal proposal that can win the support of 50 of the Senate’s 52 Republicans — a goal that has proved elusive. It would repeal the expansion of Medicaid under the Affordable Care Act, as well as the tax credits that are provided to help people buy insurance on the individual market. In their place, it would provide block grants to the states to use for health care.