Republican Sen. Ron JohnsonRonald (Ron) Harold JohnsonGun proposal picks up GOP support GOP lawmaker to unveil bill banning gun bump stocks Senate Homeland Security chairman backs bump-stock ban after Las Vegas shootings 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 ReidChris Murphy’s profile rises with gun tragedies Republicans are headed for a disappointing end to their year in power Obama's HHS secretary could testify in Menendez trial MORE [(D-Nev.)] in the Senate, I think is next to impossible."

Several Republican senators, including Sens. Ted CruzRafael (Ted) Edward CruzWhatever you think the Alabama special election means, you’re probably wrong This week: Congress gets ball rolling on tax reform Week ahead: Senators work toward deal to fix ObamaCare markets MORE (R-Texas), Mike LeeMichael (Mike) Shumway LeeWhatever you think the Alabama special election means, you’re probably wrong How the effort to replace ObamaCare failed Overnight Regulation: Trump temporarily lifts Jones Act for Puerto Rico | Bill would exempt some banks from Dodd-Frank | Senators unveil driverless car bill MORE (R-Utah) and Marco RubioMarco Antonio RubioOvernight Defense: Tillerson, Trump deny report of rift | Tillerson says he never considered resigning | Trump expresses 'total confidence' in secretary | Rubio asks Army to kick out West Point grad Rubio asks Army to kick out West Point grad with pro-communist posts GOP establishment doubts Bannon’s primary powers 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 CantorEric Cantor offering advice to end ‘immigration wars’ Trump's olive branch differs from the golden eras of bipartisanship After divisive rally, Trump calls for unity 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."