Republican leaders in the House and Senate are boxed in on immigration and searching for a way out.

Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE (R-Ky.) and 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) are struggling to find a way to fund the Department of Homeland Security while meeting conservative demands to unwind President Obama’s executive actions giving legal status to millions of immigrants who would otherwise face deportation.

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Congress is only scheduled to be in session for three weeks in February, giving lawmakers little time to craft a funding bill that would prevent an embarrassing shutdown of the Department of Homeland Security (DHS) on Feb. 28.

On Tuesday, the two began wiggling their way out of the tight spot:

• To please conservative critics, 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 announced at a morning GOP conference meeting that the House would sue the administration for giving de facto legal status to millions of illegal immigrants without congressional approval.

• A day after pulling a House border security bill drafted by Homeland Security Committee Chairman Michael McCaul (R-Texas) from a planned floor vote, GOP leaders said McCaul and Judiciary Committee Chairman Bob Goodlatte Robert (Bob) William GoodlatteNo documents? Hoping for legalization? Be wary of Joe Biden Press: Trump's final presidential pardon: himself USCIS chief Cuccinelli blames Paul Ryan for immigration inaction MORE (R-Va.) would work together to create a new, tougher border bill.

• McConnell announced at an afternoon press conference that the Senate will vote next week on a House-passed bill funding the DHS but reversing Obama’s executive actions.

• Sen. Roy Blunt Roy Dean BluntCDC tells Congress it urgently needs billion for vaccine distribution On The Money: Trump undercuts GOP, calls for bigger COVID-19 relief package | Communities of color hit hardest financially by COVID-19 | Businesses, states pass on Trump payroll tax deferral Trump undercuts GOP, calls for bigger COVID-19 relief package MORE (Mo.), the vice chairman of the Senate Republican Conference, pledged at a press conference that Senate Republicans would do “everything we can to persuade at least half a dozen Democrats that they should join us,” adding, “you don’t know how these legislative battles go if you don’t have them, and we intend to have this one.”

McConnell and Boehner have a ways to go before they are in the clear, however.

And a familiar endgame — in which Boehner is forced to move legislation through the House with broad Democratic support — could be inevitable.

The House bill is destined for failure in the Senate, where GOP leaders lack the votes to overcome a Democratic filibuster.

Almost the entire Senate Democratic caucus signed a letter Tuesday stating, “the House bill cannot pass the Senate” and urging McConnell to move a clean bill.

A lawsuit is likely to take years to wend its way through the courts, giving little satisfaction to conservative voters who want to see immediate action.

Conservatives have been divided over the McCaul bill, and it is unclear whether GOP leaders can make it more palatable by merging it with Goodlatte’s legislation, which deals with asylum claims and other issues related to enforcement.

McConnell on Tuesday declined to reveal whether he would allow Democrats to offer amendments to the House bill when it is considered in the Senate or how he would proceed if Democrats filibuster it.

Senate Majority Whip John Cornyn John CornynBipartisan praise pours in after Ginsburg's death Chamber of Commerce endorses McSally for reelection Airline job cuts loom in battleground states MORE (R-Texas) said the House bill would come directly to the Senate floor but acknowledged leaders have yet to firm up a plan should it fail.

“I would say all options are on the table. There hasn’t been any decision made, so we’re considering that full range of options,” he said.

Some of the options were discussed at a Republican Steering Committee meeting last week.

The GOP could strip the House provision on Obama’s executive actions and instead add language strengthening border security or expanding H-1B visas for high-skilled workers.

One Senate GOP source said the upper chamber could add whichever version of the McCaul-Goodlatte border security and interior enforcement bill passes the House.

Allowing Obama’s executive actions to stand and adding components of immigration reform favored by Republicans might persuade a group of centrist Democrats to join with the majority of the Senate GOP conference to pass an alternative.

But it would likely face a party-line vote and a mass of conservative defections in the House.

And winning over Democratic senators would not be easy; Sen. Charles Schumer Chuck SchumerPelosi orders Capitol flags at half-staff to honor Ginsburg Ginsburg in statement before her death said she wished not to be replaced until next president is sworn in Democrats call for NRA Foundation to be prohibited from receiving donations from federal employees MORE (N.Y.) and other members of the Senate Democratic leadership would fight efforts to pass piecemeal immigration reform on an appropriations bill.

Senate Republicans could try to sweeten the deal by adding other elements of immigration reform favored by Democrats to the Homeland Security funding bill.

That might build a bipartisan majority in both chambers but would likely prompt a harsh reaction from the GOP base.

Scott Wong and Cristina Marcos contributed.

This story was updated at 2:27 p.m. Thursday.