Republican leaders in Congress are coming under conflicting pressures as influential voices in the party debate whether the new majority should work with President Obama or oppose him at every turn.

Even before their midterm triumph, Speaker John Boehner John Andrew BoehnerLongtime House parliamentarian to step down Five things we learned from this year's primaries Bad blood between Pelosi, Meadows complicates coronavirus talks MORE (R-Ohio) and Senate GOP Leader Mitch McConnell Addison (Mitch) Mitchell McConnellOcasio-Cortez to voters: Tell McConnell 'he is playing with fire' with Ginsburg's seat McConnell locks down key GOP votes in Supreme Court fight Video shows NYC subway station renamed after Ruth Bader Ginsburg MORE (R-Ky.) had suggested a successful election would give Republicans the opportunity to show they can achieve things in Washington.

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Since their victories, the leaders have signaled they’re ready to move bills on trade and taxes to Obama’s desk, and McConnell in particular has suggested that there are several GOP-backed measures that the president could sign into law.

That’s provoking an uprising from some voices on the right, who argue that seeking compromise with Obama is the wrong strategy in the wake of Tuesday’s GOP rout.

In an editorial titled “The Governing Trap,” editors at the The National Review predicted the “prove-you-can-govern strategy” would backfire, encouraging Democrats to filibuster GOP bills while highlighting the divide between the Tea Party and establishment Republicans.

But conservative opinion is hardly unified on the issue.

Conservative columnist Charles Krauthammer wrote Thursday that the midterm elections has handed the GOP “a rare opportunity to retroactively redeem its brand” by legislating.

“Show that with Harry Reid no longer able to consign House-passed legislation to oblivion, Congress can actually work. Pass legislation,” Krauthammer wrote. “When Obama signs, you’ve shown seriousness and the ability to govern. When he vetoes, you’ve clarified the differences between party philosophies and prepared the ground for 2016.”

The emerging divide on the right is a worry for a party that has seem internal divisions sap its momentum in the past.

With Republicans set to assume control of Congress for the first time in eight years, GOP leaders must decide which policies to push forward, and whether some bigger goals should be pushed to the side.

Those taking the “don’t govern” approach argue measures to give Obama “fast-track” trade authority, tackle patent “trolls” or to repeal the healthcare law’s tax on certain medical devices are small-ball goals that should be swept aside in favor of more sweeping measures, such as repealing the individual mandate in ObamaCare.

“Passing all of the Washington lobbyists’ wish-list items is not going to get Republicans through 2016,” said Matt Kibbe, president of the Tea-Party group FreedomWorks, which has frequently criticized GOP leaders.

Kibbe urged leaders to tackle the individual mandate and reign in so-called “risk corridors” in the healthcare law that conservatives deride as bailouts for insurance companies. He also wants to see Congress vote on an alternative to the healthcare law, something Republicans failed to do this past election cycle.

Sen. Ted Cruz (R-Texas), the Tea-Party favorite who has repeatedly been a thorn in the side of GOP leadership, says his party needs to do “everything humanly possible to repeal Obamacare.”

Those arguing the GOP should send legislation to Obama’s desk, such as Krauthammer, argue it is important for Republicans to show voters that their recipes on policy can succeed and that they are serious about putting them in place.

The pro-governing side also argues that many of the House-passed GOP bills that languished in the Democratic Senate will be popular, and that if Obama vetoes them, it will play to GOP strengths.

At his press conference on Thursday, Boehner wasn’t asked about the governing debate, But Michael Steel, the Speaker’s spokesman, said in a statement that the responsibility to govern doesn’t only fall to Congress.

“Under our Constitution, one branch of the federal government cannot ‘govern’ alone,” Steel said “However, we have been clear that we will continue to listen to the American people and act on their priorities – starting with the sensible, bipartisan House-passed jobs bills like the Keystone pipeline.

“All House and Senate Republicans agree on these responsible measures,” he said.

McConnell has suggested Obama and Republicans might find common ground on trade agreements and tax reform, and suggested there’s bipartisan support in Congress for authorizing construction of the Keystone XL pipeline.

Boehner at his news conference struck a more confrontational tone with Obama — perhaps a sign of the pressure he’s facing from Tea-Party conservatives in his conference.

In Tuesday’s lopsided election, Boehner said, voters gave a ringing endorsement of GOP policies. Republicans will vote to repeal Obamacare, he insisted, saying the president should be open to bipartisan tweaks to his signature legislative achievement.

And while Boehner said it was time for Congress to deal with the prickly subject of immigration reform, Boehner said there would be hell to pay if Obama goes it alone and signs an executive order halting all deportations of illegal immigrants.

“If he acts unilaterally on his own,” Boehner said, “he will poison the well and there will be no opportunity for immigration reform moving forward in this Congress.”

Out on the campaign trail this fall, Sen. John McCain (R-Ariz.) said he repeatedly heard cries for cooperation from voters.

“We need a positive agenda,” McCain, the GOP’s 2008 presidential nominee, said on MSNBC Thursday. “I traveled around the country campaigning for our Senate candidates and one thing people [said] is they want us to get things done. We can find common ground.”

But Cruz, who appears ready to launch a 2016 bid and has refused to back McConnell as majority leader, recently told The Washington Post the newly empowered Senate Republicans should operate just like the combative GOP-controlled House. Senate committees should hold a series of hearings investigating what he sees as Obama’s abuse of power when it comes to the IRS, Benghazi and other controversies.

And at an Election Night victory party in Texas, Cruz said Republicans need to do everything they can to repeal ObamaCare, even if a bill faces no chance of getting past Democrats or a presidential veto.

“Now is the time to go after and do everything humanly possible to repeal ObamaCare. Now is the time to stand up to the president and say no more amnesty,” Cruz said. “Now is the time to stand up and say follow the Constitution, honor the rule of law and protect the Bill of Rights.”