WASHINGTON — Mitch McConnell of Kentucky will have his first big test as U.S. Senate majority leader next week as funding for the Department of Homeland Security is set to lapse Feb. 27, triggering a shutdown of some of the agency’s operations unless Congress acts.

McConnell and House Speaker John Boehner of Ohio have been trying to advance an agency funding measure that would force President Obama to reverse his orders easing deportations of undocumented immigrants.

Aside from backing down on their demand, or allowing funding to expire, there’s a third option under consideration that could take pressure off McConnell and Boehner. Lawmakers may vote on 30-day funding for the agency, according to a congressional aide who didn’t want to be identified because talks are continuing.

The congressional debate will focus attention on Obama’s directives on immigration that would ease deportation for about 5 million undocumented immigrants including those brought to the U.S. as children. A Texas judge’s order this week forced the White House to delay carrying out its immigration orders.

The short-term funding legislation would allow a White House appeal of the judge’s decision to play out. The Justice Department plans to seek a stay of the order, White House press secretary Josh Earnest said Friday.

“If the appeals court reverses the lower court judge, Republican leaders will likely turn to their colleagues and say, ‘There’s nothing we can do,’ ” said Keith Appell, a Republican strategist. “If it upholds the lower court judge, then everybody does a happy dance. Either way, the Republican leadership may be off the hook.”

On Monday, McConnell will try for a fourth time to advance a House-passed funding bill that would force Obama to abandon the immigration action he announced in November.

Democrats have already blocked the measure three times, and they’re expected to do so again.

“Republicans are fully in charge of Congress, and amid very real terrorist threats, they should pass a clean bill to fully fund our Homeland Security, free of controversial policy riders,” said Sarah Feldman, a spokeswoman for Sen. Claire McCaskill, a Missouri Democrat who has been critical of the president’s policy.

McConnell needs at least six Democrats to get the 60 votes needed to advance the bill. If the bill doesn’t move forward, Republican leaders must decide whether to give up their fight, something that would spark intense criticism from conservative lawmakers in Congress.