WASHINGTON — When it comes to managing Republicans’ best interests, Senator Mitch McConnell of Kentucky, the majority leader, rarely loses. So it is possible that Mr. McConnell views the potential failure of a hastily written health care bill as an eventual boon.

His presentation on Thursday of the Senate’s health care measure to Republican colleagues — after the White House and key lobbyists got a peek the night before — was met with something other than unbridled enthusiasm. According to lawmakers who were at the unveiling, members from the left and right ends of the party’s spectrum were deeply critical of the effort.

As Democrats immediately took to the Senate floor to excoriate the bill and the secretive process in which it was put together, few Republicans, even those involved in crafting it, came to defend it.

The Run-Up

The podcast that makes sense of the most delirious stretch of the 2016 campaign.

A handful of Republicans — more than Mr. McConnell can afford to lose — were quick to disparage the measure. “I have serious concerns about the bill’s impact on the Nevadans who depend on Medicaid,” Senator Dean Heller, his party’s most vulnerable incumbent in the 2018 elections, said of his constituents.

Senator Susan Collins of Maine rendered her own lukewarm judgment, while Senator Shelley Moore Capito of West Virginia simply said she would read it, with all the enthusiasm of a college senior faced with a weekend assignment of Proust.

Four others went further. Senators Ted Cruz of Texas, Ron Johnson of Wisconsin, Mike Lee of Utah and Rand Paul of Kentucky all said they would not vote for the bill as currently proposed.

Mr. McConnell plays his strategic cards so close to the vest that a Queen of Hearts must be tattooed on his tie. He may, of course, be convinced that the Senate can pass this bill. Perhaps after some moaning, and some changes to the bill through amendments, the 51 senators needed to get the bill over the line (or 50 if Vice President Mike Pence is summoned) will choose a good-enough effort over being tarred as the person who declined to make good on a seven-year promise to unravel President Barack Obama’s signature domestic policy achievement.

When it comes to voting yes, a majority of members of Congress have a policy price, and leaders often will write the check. “There’s the natural frustrations that people have” at the start of a process that often ends in legislative victory, said Senator Bob Corker, Republican of Tennessee.

But there are potential costs for senators like Mr. Heller and others in repealing a law that has grown in popularity over recent years, and Mr. McConnell has always taken pride in protecting his members. Trying to come to a meeting of the minds with the House — which crafted a far more conservative bill in many respects — would be time-consuming and unpleasant.

Mr. McConnell and many of his aides are also eager to get to the business of changing the tax code, which they view as less difficult than health care, and have been working with the White House behind the scenes to get that effort started. For Mr. McConnell, cutting taxes is a much higher priority than health care, which time and President Trump have turned into quicksand for him and his fellow Republicans.

Just hours after the presentation of the Senate health care bill, Mr. McConnell met with Speaker Paul D. Ryan and White House officials to talk taxes.

Mr. McConnell is not fond of bringing bills to the floor that he does not think can pass. Should he be unable to pull together enough support on the health care bill over the next week, it would seem likely at first glance that he would make the dreaded call to the White House to let the president know that he lacked the votes.

When Mr. Ryan made that move, it was received with anger and pressure to get something — anything — off the floor, and indeed Mr. Ryan did. But Mr. McConnell and senators are generally more resistant to pressure from the White House and will keep their own interests in mind. Forcing senators from states where the Medicaid law was expanded to take that vote and have the bill fail could be costly.

“It’s a short bill,” said Mr. Corker, who, like most other members, said he still had to plow through it and talk to state insurance officials, among other steps. “But it has a big impact on a lot of people.”

However, Mr. McConnell may also decide that the matter cannot be closed without a vote and take his chances that recalcitrant members can be pulled along. Not voting would also leave House Republicans, who voted for a deeply unpopular bill in their chamber, exposed.

Simply put, coming up with a final version of the bill that pleases Ms. Collins, who is concerned that it is still too punitive for many residents of her old and relatively poor state, and Mr. Paul, who feels it is still too generous, is a tough task.

“I think everybody wants to get to yes,” said Senator John Thune of South Dakota, a member of the Republican leadership. “And there are some things that we’ve said all along that are dialable on this bill that we can hopefully tweak a little bit before it comes to the floor.”

He added, however, “I’m not sure that, you know, Rand will ever be there.”

Many Republicans say privately that they are eager to work with Democrats on fixes to the current law to keep the insurance exchanges from imploding. They may well wish to call Democrats’ bluff as they have insisted they want that to work in a bipartisan fashion, too.

But most instructive of all may be Mr. McConnell’s own words. In his 2016 memoir, “The Long Game,” he noted that, as minority leader, he went out of his way to make sure that one party owned the health care issue. “I wanted a clear line of demarcation — they were for this, and we were against it,” he said. Perhaps he is not excited to let that one party now be his own.