Sens. John McCain, Lindsey Graham, and Ron Johnson AP The Senate Republican healthcare process descended into disarray Thursday evening, as four GOP senators threatened to withhold support for a more moderate, "skinnier" attempt at repealing certain parts of the Affordable Care Act.

"I am not going to vote for the skinny bill if I'm not assured by the House there will be a conference where my idea and other ideas will be taken up so we can actually repeal Obamacare," Sen. Lindsey Graham of South Carolina said in a press conference Thursday with Sens. John McCain, Ron Johnson, and Bill Cassidy.

Graham added: "I'm not going to vote for a bill that is terrible policy and horrible politics just because we have to get something done."

The last-ditch, so-called skinny repeal effort would consist of a series of amendments would aim to repeal certain unpopular parts of the Affordable Care Act, including mandates to get health insurance.

At the end of the 20-hour debate period, if no bill has been picked up, the Senate Majority Leader Mitch McConnell would move on to the "skinny repeal" plan. If passed, it could lead to the House and Senate working together to compromise on one final bill in conference.

The aim is to find common ground with the House on a bill that becomes more fleshed-out. But the House doesn't necessarily have to call a conference on the bill. It could simply pass the bill and send it to the president.

After the press conference, House Speaker Paul Ryan expressed a willingness to take the bill to conference.

"If moving forward requires a conference committee, that is something the House is willing to do," he said.

The four GOP senators said they wanted assurances from House Republican leadership — including House Speaker Paul Ryan and Majority Leader Kevin McCarthy — that they would not pass the bill, which Graham called a "fraud."

The senators didn't specify what kind of assurances they would request.

"It's like pornography. You'll know it when you see it," Graham said.

The House is supposed to go into recess starting on Friday, but on Thursday, they were told to stay flexible with their travel plans.

"While last votes are currently scheduled to take place tomorrow, Members are advised that — pending Senate action on healthcare — the House schedule is subject to change," McCarthy wrote. "All Members should remain flexible in their travel plans over the next few days."

What 'skinny repeal' would look like:

The "skinny repeal" bill would repeal both the individual and employer mandates, which requires individuals to have health insurance, and employers to provide health insurance to employees. If they don't, they face a penalty fee under Obamacare.

It would also repeal some of the taxes that the ACA put in place — most significantly, a tax on medical-device makers.

The skinny bill would likely leave everything else untouched.

The bill could also have some language around waiving essential health benefits, which could require the bill to get 60 votes to pass without a filibuster.

While the text of the "skinny repeal" bill hasn't been released yet, the bill would be nowhere near as extensive as the full-repeal plan or the BCRA. But it would seek to alter the much-criticized mandates that Republicans have targeted for years.

The Congressional Budget Office scored the "skinny" plan on Wednesday and found that it would leave 16 million more Americans without health insurance.