Speaker Paul Ryan Paul Davis RyanKenosha will be a good bellwether in 2020 At indoor rally, Pence says election runs through Wisconsin Juan Williams: Breaking down the debates MORE (R-Wis.) said the House is willing to go to conference with the Senate on a healthcare bill the upper chamber is expected to approve Thursday night.

Ryan offered the assurance after Sen. John McCain John Sidney McCainBiden's six best bets in 2016 Trump states Replacing Justice Ginsburg could depend on Arizona's next senator The Hill's Morning Report - Sponsored by Facebook - Washington on edge amid SCOTUS vacancy MORE (Ariz.) and three other Republicans said they wanted a guarantee the House would not seek to just pass the scaled-back Senate bill instead of doing a conference, which could result in the emergence of a broader bill.

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“Senators have made clear that this is an effort to keep the process alive, not to make law,” Ryan said in a statement. “If moving forward requires a conference committee, that is something the House is willing to do.”

Ryan also said he wanted assurances that the Senate would vote first on a conference report. This would ensure the Senate would have to take a potentially tough vote before the House.

“The House remains committed to finding a solution and working with our Senate colleagues, but the burden remains on the Senate to demonstrate that it is capable of passing something that keeps our promise, as the House has already done,” Ryan said in his statement.

“Until the Senate can do that, we will never be able to develop a conference report that becomes law. We expect the Senate to act first on whatever the conference committee produces.”

However, McCain indicated that Ryan's statement didn't go far enough to explicitly rule out the House voting on the "skinny" repeal bill at some point.

"I would like to have the kind of assurances he did not provide," McCain said.

GOP leaders in the Senate have sought to win support for the scaled-back bill by saying it is a way to get to a conference with the Senate.

But moves by the House had raised questions about whether the lower chamber might just approve the Senate bill.

The House approved special rules that would have allowed early consideration of legislation through the beginning of next week. GOP leaders in the House also advised members that the beginning of the House recess could be adjusted if the Senate approved its bill.

Democrats in the Senate pounced on those moves, arguing they suggested that the House GOP planned to move forward on the Senate "skinny bill."

After Ryan's statement, Sen. Brian Schatz (D-Hawaii) in comments on the Senate floor warned Republicans that Ryan's statement just amounted to a lot of words. Sen. Tim Kaine Timothy (Tim) Michael KaineTrump meets with potential Supreme Court pick Amy Coney Barrett at White House Names to watch as Trump picks Ginsburg replacement on Supreme Court Barrett seen as a front-runner for Trump Supreme Court pick MORE (D-Va.) also argued that if the scaled-back bill were approved, the House would likely approve it.

The scaled-back bill the Senate is expected to pass Thursday night will likely include a repeal of ObamaCare's individual insurance mandate and a partial repeal of the employer mandate.

It would also defund Planned Parenthood for a year.

Before Ryan’s remarks, McCain and Sens. Lindsey Graham Lindsey Olin GrahamGraham: GOP will confirm Trump's Supreme Court nominee before the election Fox's Napolitano: Supreme Court confirmation hearings will be 'World War III of political battles' Grassley, Ernst pledge to 'evaluate' Trump's Supreme Court nominee MORE (R-S.C.), Bill Cassidy (R-La.) and Ron Johnson Ronald (Ron) Harold JohnsonThis week: Supreme Court fight over Ginsburg's seat upends Congress's agenda GOP set to release controversial Biden report Democrats fear Russia interference could spoil bid to retake Senate MORE (R-Wis.) held a press conference to make it clear that they had problems with the "skinny bill" and that they thought it would be bad policy if it became law.

They acknowledged worries the House might just act on the bill.

“There's increasing concern on my part and others that what the House will do is take whatever we pass” and pass it without making changes, Graham said.

Updated: 9 p.m.