Mr. Cassidy said Vice President Mike Pence was encouraging senators and governors to support the proposal for block grants, which he drafted with Senator Lindsey Graham, Republican of South Carolina.

Kellyanne Conway, a top adviser to Mr. Trump, said on Fox News: “The president is ready. He’s ready with pen in hand to sign health care reform if, say, Graham-Cassidy moves forward. A lot of the governors seem to be supportive of that.”

Describing his proposal, Mr. Cassidy said: “Every American, no matter where that American lives, will ultimately get about the same amount of money to support health care. Right now four states get 37 percent of all the money that is spent on behalf of the Affordable Care Act. We equalize that.”

Mr. Cassidy said his bill was being drafted so it could be approved in the Senate with a simple majority, rather than the 60 votes usually required for major legislation.

But Mr. Cassidy and his bill face long odds. The procedural device that allows Republicans to sidestep a Democratic filibuster on health care will not be available after Sept. 30, little more than three weeks from now, the Senate parliamentarian said last week.

Mr. Cassidy has not finished writing the bill, no committee has held a hearing on it, and the Congressional Budget Office would need to analyze it — a task that could take two weeks — before the Senate could vote on it.

Optima Health, a Virginia-based insurer, announced on Wednesday that it would limit its participation in the individual insurance market next year, raising the possibility that people in dozens of Virginia counties might not have any insurance options. Senator Tim Kaine, Democrat of Virginia, said the announcement showed the urgent need for congressional action and the uncertainty caused by the Trump administration’s threat to stop the cost-sharing subsidies.