Republican Sen. Ron JohnsonRon JohnsonOvernight Healthcare: Senate votes to begin ObamaCare repeal debate | McCain returns to vote | GOP floats scaled-down healthcare bill MSNBC ‘Cartoon of the Day’ shows McCain lying in hospital bed McCain returns to applause, casts deciding vote to advance healthcare bill MORE on Tuesday said it would be "next to impossible" to use a government funding bill to stop ObamaCare.

"Even if we were to not pass the continuing resolution [to fund the federal government], you're not going to be able to defund Obamacare, absent of President Obama signing a law, which I think is highly unlikely," the Wisconsin senator told the Milwaukee Journal Sentinel.

"So I appreciate the fact that they've raised the issue. But defunding Obamacare, with President Obama in the White House and Harry ReidHarry ReidConservative Senate candidate calls on GOP to end filibuster Ex-Reid aide: McConnell's 'original sin' was casting ObamaCare as 'partisan, socialist takeover' GOP faces growing demographic nightmare in West MORE [(D-Nev.)] in the Senate, I think is next to impossible."

Several Republican senators, including Sens. Ted CruzTed CruzSenate rejects ObamaCare repeal, replacement amendment Senate Dem: We’re trying to block a recess appointment to replace Sessions Sessions says he won't resign despite Trump criticism: report MORE (R-Texas), Mike LeeMike LeeSenate rejects ObamaCare repeal, replacement amendment Trump turns up heat on AG Sessions over recusal GOP senators back Sessions after Trump attacks MORE (R-Utah) and Marco RubioMarco RubioMexican politicians have a new piñata: Donald Trump Bush ethics lawyer: Congress must tell Trump not to fire Mueller The private alternative to the National Flood Insurance Program  MORE (R-Fla.), are putting pressure on GOP leaders to risk a government shutdown if Democrats don't agree to repeal the healthcare law.

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“If you fund this thing, you own it,” Lee said Tuesday on the "Sean Hannity" radio show.

“We cave and we cave and we cave,” he added. “This is how we get into trouble.”

But other top Republicans, including House Majority Leader Eric CantorEric CantorSpecial interests hide behind vets on Independence Day What to watch for in Comey’s testimony Trump nominates two new DOD officials MORE (R-Va.), have urged Republicans to abandon that hope, noting that there are not enough votes in the Senate.

"In order to avoid a government shutdown, we need 60 votes in the Senate and 218 votes in the House to pass a continuing resolution," Cantor told National Review's Robert Costa. "To get 60 votes in the Senate, you need at least 14 Democrats to join Republicans and pass a CR that defunds Obamacare. Right now, I am not aware of a single Democrat in the Senate who would join us. If and when defunding has 60 votes in the Senate, we will absolutely deliver more than 218 votes in the House."

At a press conference last week, President Obama said he assumed Republicans would not attempt to force the shutdown.

"The idea that you would shut down the government at a time when the recovery is getting some traction; where we’re growing, although not as fast as we need to; where the housing market is recovering, although not as fast as we would like; that we would precipitate another crisis here in Washington that no economist thinks is a good idea — I’m assuming that they will not take that path," Obama said.

"I have confidence that common sense, in the end, will prevail."