Republicans have 52 senators, and a tie vote on the new bill would have to be broken by Vice President Mike Pence, a fellow Republican.
And right now, there are more expected "no" votes than just two.
At the same time, McConnell, R-Ky., has said he wants to have a vote on the bill by late next week, before Congress' Fourth of July recess. This would let senators go home for several weeks without having to hear concerns about the bill from their constituents in person.
Robert Laszewski, president of the consulting firm Health Policy and Strategy Associates, in a client update issued Thursday, wrote, "If McConnell and the Republicans can" win passage of the bill next week, "it will be one of the most masterful legislative jobs in recent memory as the Republican leader."
McConnell, Laszewski wrote, will try "to thread a needle balancing the demands of conservative Republicans trying to unwind the 'Democratic welfare state,' and a handful of more pragmatic moderate Republicans opposed to simply throwing millions of people off their coverage."
Hours after the 142-page bill was released, four GOP senators, Rand Paul of Kentucky, Ted Cruz of Texas, Ron Johnson of Wisconsin, and Mike Lee of Utah, gave a sense of how tight that needle is when they issued a joint statement saying "we are not ready to vote for this bill."
That quartet of conservatives said they "are open to negotiation and obtaining more information" before a vote. But they added that the draft does not appear that it will fulfill what they called "the most important promise that we made to Americans: to repeal Obamacare and lower their healthcare costs," their statement said.
Cruz said later, "This current draft doesn't get the job done." But he added, "I think we can get to yes," particularly if the bill can guarantee a reduction in insurance premiums.
Later in the day, Sen. Susan Collins, R-Maine, when asked in an MSNBC interview what "line in the sand" she would not cross to support the bill, said, "I cannot support a bill that's going to greatly increase premiums for older Americans or out-of-pocket costs for those who aren't quite old enough for Medicare yet."
"I cannot support a bill that's going to result in millions of people losing their health insurance," said Collins, a moderate.
"And I cannot support a bill that is going to make such deep cuts to Medicaid that's going to shift billions of dollars in costs to our state government, to those who have insurance, and to health-care providers such as rural hospitals which would be faced with a great deal of uncompensated care," Collins said.
NBC News' Chuck Todd, referring to the fact that the bill, at the moment, is likely to do just that, asked Collins if she could imagine passing enough amendments to the bill to assuage her concerns. The senator replied, "I am sure many of us are going to have amendments."
Two other moderates separately released statements of their own.
Sen. Dean Heller, R-Nev.,said he has "serious concerns about the bill's impact on Nevadans who depend on Medicaid."
"I want to make sure the rug is not pulled out from Nevada or the more than 200,000 Nevadans who received insurance for the first time under Medicaid expansion," Heller said.
Sen. Shelley Moore Capito, R-W.Va., in her own statement, said she will weigh whether the bill "provides access to affordable health care for West Virginians, including those on the Medicaid expansion and those struggling with drug addiction."
Other Republican senators are likely to have concerns both about the phaseout of Medicaid's expansion under Obamacare and about a cap on spending for Medicaid overall, both of which are called for by the bill. There also will be concerns that the bill's restructuring of the system for subsidizing customers of individual health plans will lead to much higher costs for many older Americans and those with health problems.