Lawmakers from both parties have spoken out in support of the proposal, and Mr. Trump had initially encouraged the talks between Mr. Alexander and Ms. Murray, the top two senators on the health committee. But on Wednesday, he wrote on Twitter that while he was supportive of the effort, “I can never support bailing out” insurance companies, which he said had made a fortune under the health law.
Sarah Huckabee Sanders, the White House press secretary, then suggested that Mr. Trump could be won over if changes were made to the legislation.
“He wants to lower premiums,” Ms. Sanders said. “He wants to provide greater flexibility. He wants to drive competition. He likes the idea of block grants to states.”
Democrats have strenuously opposed block grants and have resisted another proposal renewed by the White House in recent days, to repeal the requirement for most Americans to have health insurance.
Senator Chuck Schumer of New York, the Democratic leader, appeared reluctant to restart negotiations. Asked on NBC’s “Meet the Press” on Sunday if Democrats would return to the negotiating table, Mr. Schumer said: “We have an agreement. We want to stick by it.”
Responding to Mr. McConnell, Mr. Schumer later said that Mr. Trump now “holds the key” to preventing premiums from rising.
“Now that Leader McConnell has made it clear he will put the Murray-Alexander bill on the floor as soon as the president supports it, the president should say that he does,” Mr. Schumer said, adding that the bill could pass if given a chance.
The proposal would probably face tougher odds in the House.