Lee was among four conservatives who announced last week that they were against the current version of the legislation.
A Congressional Budget Office score on Monday estimated that the proposal would lead to 22 million more uninsured Americans by 2026, only complicating matters for moderate GOP senators on the fence.
McConnell has a narrow majority of 52 seats in the Senate.
McConnell and a small group of senators had worked in secret for weeks on a bill passed by the House.
Trump spoke Tuesday at a gathering of Senate Republicans after Senate leaders shelved the vote.
There were some in the GOP who were not pleased with the decision to delay; Sen. Capito and another moderate, Sen.
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But both Republican bills threaten to make it more hard for lower-income Americans to access healthcare, as well as curtail access to services such as Planned Parenthood and programmes that tackle drug addiction, according to Democrats and campaigners.
Our job now is to make sure that 100% of the American people know what's in the disastrous Republican "health care" bill. Susan Collins (ME), Rand Paul (KY), Dean Heller (NV), Ron Johnson (WI), and Mike Lee (UT) - publicly said they would oppose a "motion to proceed" vote required to bring the bill up for consideration. She's always been a question mark on how she'll vote on the bill, and she told CNN Tuesday she was "concerned about the bill in the form that it is now".
"If we're going to reject something that is offered other than Obamacare, then you are saying you want Obamacare over that", Inhofe told reporters after leaving the closed door GOP meeting. "Either Republicans will agree and change the status quo, or markets will continue to collapse and we'll have to sit down with Sen".
Mr Trump has called the House bill "mean" and prodded senators to produce a package with more "heart".
"The Senate healthcare bill missed the mark for Kansans and therefore did not have my support", Moran, who is in his first term, said in a statement posted on Twitter. Should the Senate pass the bill, Ryan argued, the House will get their chance, one way or another, to weigh in.
It would however cut deeply into Medicaid, America's public health programme for the poor and disabled.
Moderate senators are anxious millions of people will lose their insurance.
Mr McConnell had hoped to push the measure through his chamber before an Independence Day recess that party leaders fear will be used by foes of the legislation to tear away support. That's because standard policies would be skimpier than now offered under Obama's law, covering a smaller share of expected medical costs.
The news followed a bustling morning in the Capitol, where Republican leaders, along with Pence, were meeting behind closed doors with the bill's opponents, as a number of senators came out against voting for a procedural step to advance their plans to repeal and replace Obamacare. Net premiums for midlevel silver plans that most consumers now buy would go up 74 percent on average under the Senate bill, from $197 a month to $342.