Senate leaders on Tuesday canceled plans to vote on an Obamacare repeal this week, marking a third straight failure for Majority Leader Mitch McConnell and fellow Republicans who have been unable to fulfill their campaign promise.

Facing too many defections, Mr. McConnell concluded it didn’t make sense to move forward, and said the chamber will now turn to other business such as a massive tax code overhaul.

The failure leaves Obamacare intact, and Democrats begged the GOP to try for bipartisan fixes to prop up the struggling law. The Republican chairman of the health committee said he’ll try to restart negotiations.

Most in the GOP, though, were cool on the idea, saying the law is hopelessly flawed.

“I think there are a number of them that are going to be very hesitant to try and put more money into a system that they are convinced is going south,” said Sen. Mike Rounds, South Dakota Republican.

Without action, it will be up to President Trump to decide how aggressively to support the law as signups for 2018 begin.

Republicans had been racing an end-of-month deadline to approve a repeal bill under the 2017 budget, which allowed them to avoid a Democratic filibuster.

Some in the GOP insisted they can try again, either by writing a 2018 budget to accommodate repeal, or including it in the 2019 budget due early next year.

But the 2018 budget is already being eyed to push through a tax reform package, and some Republicans were wary of loading it down with several big lifts.

“The problem is, I don’t know that we can keep them separate, just to be frank,” Rep. Mark Meadows, North Carolina Republican and chairman of the archconservative House Freedom Caucus, told reporters at a Conversations with Conservatives event moderated by the Heritage Foundation.

Rep. Thomas Massie, though, said Democrats packed multiple issues into the budget they used to help pass the Affordable Care Act originally, and said it could be done for repeal.

“You could absolutely do two things in one,” the Kentucky Republican said. “It’s not dead on September 30th. Like many things here, it’s a fake deadline.”

The House passed an Obamacare repeal bill in May, but the Senate has now failed three times.

This latest attempt came on a bill from Sens. Lindsey Graham of South Carolina and Bill Cassidy of Louisiana. Under the plan, money that funded Obamacare’s expansion of Medicaid and subsidized many of the people who bought plans on its exchanges would be pooled together and siphoned back to the states.

Yet Sen. Susan Collins of Maine late Monday said she opposed the plan, citing cuts to Medicaid coverage and estimates that millions would lose coverage. Republican Sens. Rand Paul of Kentucky and John McCain of Arizona had already defected, forcing sponsors to admit they’d run out of time this week.

Liberal activists cheered the GOP’s failure, while Democrats pleaded for a new start to rescue the law.

“Instead of trying to cut health care, we Democrats are ready, willing and eager to work to improve our health care system,” said Senate Minority Leader Charles E. Schumer. He said health committee Chairman Lamar Alexander and Sen. Patty Murray, the ranking Democrat, were “very close to an agreement.”

Republicans, though, still have their eye on repeal.

“We’re coming back to this after taxes,” Mr. Graham insisted. “We’re going to take this show on the road.”

In recent days, he’d said that if repeal failed now, he would demand it be part of the 2018 budget, and wouldn’t support a blueprint without it. By Tuesday, though, his stance appeared to soften.

“I’ve been told by people it makes it extremely complicated, and what I want to do is take health care up in the most advantageous way, to understand what we’re doing,” he said.

Sen. Ron Johnson, Wisconsin Republican, said Tuesday it sounds like GOP leadership might have already ruled out fusing the two issues in the 2018 budget, but that he’d still like to try to include health care.

“I have not ruled that out - I want to talk to leadership,” he said. “But I sure don’t want to jeopardize tax reform, either.”

“Listen, I’m [a] pretty flexible guy,” he said when asked about using the 2019 budget. “I’ll accept reality. I don’t particularly like the current reality, but it is what it is.”

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