A short-lived push by Sen. Lindsey O. Graham (R-S.C.) to drag the GOP health-care fight into next year fell apart after Senate leaders warned that demands for health-care votes on an upcoming budget bill would cripple plans to pass a GOP tax bill by the end of this year.

Graham said Tuesday that he supports turning full attention to passing a Republican tax plan and delaying any further health bill until next year. The announcement was part of a sweeping defeat of a last-ditch GOP effort to repeal and replace the Obama-era Affordable Care Act this week.

“I’ve been told by people that it makes it extremely complicated,” Graham said. “What I want is to take health care up in the most advantageous way. So patience is a virtue, and time is actually on our side.”

Graham’s announcement marked a rapid retreat from his earlier demands that the Senate vote this week on a health-care bill he co-authored with Sen. Bill Cassidy (R-La.) or agree to pairing health care and taxes in an upcoming budget bill. But that plan collapsed after Senate Majority Leader Mitch McConnell (R-Ky.) and his deputies insisted that pairing health care with taxes would lead to certain defeat on both issues.

“We’re not going to do that,” said Senate Finance Committee Chairman Orrin G. Hatch (R-Utah). “It would screw up the budget.”

Republican leaders have been eager to move on from their repeated failures on health care in hopes of passing a tax bill that will demonstrate their ability to govern. McConnell and other GOP leaders believe that the 52 Republican senators can unite over taxes and salvage a year that is absent any major legislative victories for their side. They plan to attach the tax measures to the budget bill, which under special Senate rules could be passed by the Republican majority without help or input from the Democrats.

Early in the week, Graham joined Sens. Rand Paul (R-Ky.) and Ron Johnson (R-Wis.) to insist that GOP dreams of cutting taxes shouldn’t be used as an excuse to abandon Republicans’ long-standing pledge to gut former president Barack Obama’s signature health-care law.

“There’s no reason you couldn’t do health care and taxes at the same time,” Paul said Monday. “You can do it every year. You’ll have at least one shot every year that Republicans control the Senate.”

But Republican leaders insisted that the tactic would harm both goals. Senate Majority Whip John Cornyn (R-Tex.) said that any attempt to combine health care and taxes on the next budget risks tainting tax discussions with old divisions over health care.

“I wouldn’t support that,” Cornyn said Monday.” I think we should give health care our shot and then move on to taxes. I don’t think we should mix those up.”

The argument appeared to sway most Senate Republicans even as the idea gained support among members of the conservative House Freedom Caucus. At a roundtable with reporters hosted by the Heritage Foundation, several Freedom Caucus members argued that there might be a way to include health care in the 2018 budget reconciliation — though none knew how, exactly.

“September 30th is a fake deadline,” argued Rep. Thomas Massie (R-Ky.), who regularly votes against the leadership. “Absolutely false. You can do it on the next reconciliation vehicle. People say you can only do one thing at a time in reconciliation? Go back and look at the Obamacare reconciliation bill.”

Rep. Mark Meadows (R-N.C.), the chairman of the caucus, said he wanted to see whether health care could be added to the package without jeopardizing the entire tax effort.

“We’d be in favor of including them in the 2018 reconciliation instructions if you can keep them separate,” Meadows said. “Since we have fumbled at least twice on health care, to include it and make tax reform contingent on us passing health care, I wouldn’t be in favor of that.”

After the House GOP’s conference meeting, Rep. Diane Black (R-Tenn.), chairman of the Budget Committee, said that adding health care to the next package was “up for consideration,” but she wouldn’t say how seriously.

Keeping the health-care effort alive is appealing to a wide swath of Republicans in Congress who have been assailed by voters for failing to repeal Obamacare. But the party has been unable to agree on an acceptable alternative, and many Republicans worry that they are hopelessly divided on the issue. Many fear that continuing to fight over health care could permanently damage the party by exposing internal differences that could imperil other legislative priorities, particularly taxes.

“I’m going to always be in favor of anything we can do to repeal the major parts of Obamacare, and if somebody slips it into tax reform, sure, I’d vote for it,” said Rep. Chris Collins (R-N.Y.) “It could possibly hurt a Republican-only tax reform [bill]. We couldn’t get the votes we needed on Obamacare.”

Mike DeBonis and Sean Sullivan contributed to this report.