“Marco has to be very careful about this,” said a Senate GOP source close to McConnell, referring to Sen. Marco Rubio (R-Fla.). “I know he’s worried about [Ted] Cruz saying this is bill is too Washington, but at the end of the day, you’ve been talking all this time about repealing ObamaCare, and this is your chance to do it.”

Rubio and Sen. Ted Cruz (R-Texas), who are both running for president, issued a joint statement last month vowing to oppose the House-passed ObamaCare repeal bill because it does not roll back the entire law. Conservative Sen. Mike Lee (R-Utah) also signed it. 

Only seven House Republicans voted against that repeal package, and the Senate version is likely to be very similar, several Senate Republicans said.

“We’re going to repeal as much of ObamaCare as possible but the House bill got an overwhelming majority, most of the Freedom Caucus voted for it, so I think we’ll pass something similar,” said a Senate Republican who requested anonymity.