Senator John Cornyn of Texas, the No. 2 Senate Republican, said there would be “endless amendments” if the procedural hurdle were cleared. He played down the significance of which substitute measure would come first.

“Everybody will get a vote on everything they want to vote on,” Mr. Cornyn said. He added, “What we’re trying to do is convince everybody that if they’d like to get a vote on their amendment, then they need to vote to proceed to the House bill.”

Democrats were incredulous.

“We are potentially one or two days away from a vote on a bill that would reorganize one-sixth of the American economy, impacting tens of millions of American lives, and no one knows what it is,” said Senator Chuck Schumer of New York, the Democratic leader. “It’s sort of like Alice in Wonderland around here.”

What they will vote on will not matter if senators oppose beginning debate. Mr. McConnell can lose only two Senate Republicans, now that Mr. McCain intends to be in the chamber.

Senator Susan Collins, Republican of Maine, is all but certain to vote no on the procedural vote, no matter what legislation Mr. McConnell promises to put before the chamber if the initial hurdle is cleared.

At least two other Republicans, Shelley Moore Capito of West Virginia and Lisa Murkowski of Alaska, have indicated they will not vote to proceed if Senate leaders plan to then put forth a measure to repeal the health law without providing a replacement.

While in West Virginia later on Monday, addressing the National Scout Jamboree, Mr. Trump teased the health and human services secretary, Tom Price, about whether he would be able to wrangle support from Ms. Capito and other Republicans. “He better get them,” Mr. Trump said, smiling at Mr. Price to indicate he was joking — or at least seemed to be. “Otherwise, I’ll say, ‘Tom, you’re fired.’”