Republican Sen. Ron JohnsonRonald (Ron) Harold JohnsonRepublicans fear backlash over Trump's threatened veto on Confederate names GOP senators debate replacing Columbus Day with Juneteenth as a federal holiday Senate Republicans defend Trump's response on Russian bounties 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 Mason ReidMcConnell warns Democrats not to change filibuster rule Filibuster reform gains steam with Democrats The Hill's Morning Report - Trump wants executive order on policing; silent on pending bills MORE [(D-Nev.)] in the Senate, I think is next to impossible."

Several Republican senators, including Sens. Ted CruzRafael (Ted) Edward CruzTrump administration grants funding extension for Texas testing sites Hillicon Valley: Democrats introduce bill banning federal government use of facial recognition tech | House lawmakers roll out legislation to establish national cyber director | Top federal IT official to step down GOP lawmakers join social media app billed as alternative to Big Tech MORE (R-Texas), Mike LeeMichael (Mike) Shumway LeeSenate panel votes 21-1 to back Justice IG measure over Graham objections Senators offer bill to expand charitable giving tax break Overnight Energy: Senate passes major lands conservation bill | Mnuchin ordered to give Native American tribes full stimulus funding | Key Republican jeopardizes Trump consumer safety nominee MORE (R-Utah) and Marco RubioMarco Antonio RubioTrump administration eyes new strategy on COVID-19 tests ACLU calls on Congress to approve COVID-19 testing for immigrants Republicans fear backlash over Trump's threatened veto on Confederate names 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 Ivan CantorTrump taps pollster to push back on surveys showing Biden with double-digit lead Bottom Line The Democrats' strategy conundrum: a 'movement' or a coalition? 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."