Ohio Gov. John Kasich said Monday that SenateGOP leaders plowing ahead with a partisan health care plan are making a mistake and should focus on bipartisan fixes to Obamacare’s “failings.”

Mr. Kasich, a Republican who ran against President Trump during the 2016 primary, has been a frequent critic of the GOP’s plans to replace the Affordable Care Act, particularly parts that phase out former President Barack Obama’s expansion of Medicaid insurance for the poor and then further curtail federal spending on the program.

“The American people will come out on the losing end if Senate Republicans try to force through a new health care proposal with no bipartisanship, transparency or open dialogue,” Mr. Kasich said in a prepared statement. “Until Congress can step back from political gamesmanship and come together with a workable bipartisan plan, it is a mistake for the Senate to proceed with a vote on Tuesday and force a one-sided deal that the American people are clearly against.”

Senate leaders are hurtling ahead with their plan to take up the House-passed health bill and debate changes, even though Majority Leader Mitch McConnell hasn’t explained which path he will take if he can gather 50 votes to proceed.

Some Republicans want to replace the Affordable Care Act now, while others want to revive a 2015 plan that would gut the law within two years, buying time for a replacement and forcing senators to explain why a plan that President Obama vetoed isn’t good enough for Mr. Trump’s signature.

It is still doubtful whether either route can muster enough votes for final passage, yet GOP leaders say the time for waffling is over and that unhappy senators should get on the bill and push for changes on the floor.

With Sen. John McCain absent and Democrats uniformly opposed, Republican leaders cannot afford more than one defection from their 52-seat GOP majority on a motion to proceed to debate, and Sen. Susan Collins of Maine has signaled she is a firm “no.”

Sen. Rand Paul, Kentucky Republican, said Monday he will vote to proceed onto the bill if leaders decide to bring up the straight-repeal option first.

Mr. Paul strenuously opposes the replacement bill that Mr. McConnell put forward. He said it leaves too much of Obamacare in place by maintaining some of its regulations and blessing new forms of spending through refundable tax credits and stabilization funds for insurers.

“I’ve told my colleagues: I try to be part of the team. But my team promised to actually repeal ObamaCare … over and over again at election time. My team voted for the 2015 clean repeal bill and told the American people they’d do that if they got the White House,” Mr. Paul wrote in an op-ed for The Hill newspaper. “My team did not promise $200 billion insurance bailouts or continuing ObamaCare taxes and regulations. So if my team remembers what they’ve said and what they’re supposed to stand for — count me in. But if they continue not to, I’ll be a hell no.”

Republicans leaders trying to wrangle their conference received a new headache Friday, when the Senate’s main referee advised that parts of their replacement plan do not comport with fast-track budget rules and should be subject to a 60-vote threshold.

Parliamentarian Elizabeth MacDonough flagged provisions that would defund Planned Parenthood for one year over its abortion practice and bar consumers from using tax credits in the GOP plan to buy health plans that cover coverage.

She also dinged the “Buffalo bailout” — a part of the House bill that said New York State could not force counties to contribute to the Medicaid program. House leaders used the measure to gain votes from moderate Republicans in upstate New York.

Rep. Tom Reed, New York Republican, said he will vote “no” if the Senate sends back a bill that doesn’t include the provision, saying Medicaid costs should be handled at the state level instead of roping in local governments.

“We cannot waste this opportunity to right this wrong, nor continue to ignore the needs of our hardworking neighbors and friends of New York,” he said.