Among the possible Plans B’s: Republicans could pass a new bill to beef up security at the U.S.-Mexico border. They could sue to overturn Obama’s unilateral protections for millions of undocumented immigrants. Or they could pass yet another short-term DHS funding measure, giving the GOP more time to approve a strategy. Either way, Republican leaders hope to reach a deal that would allow Homeland Security funding to continue past Feb. 27, without making it appear to their right flank that they are caving to the White House.

All these options would soften the hard line the House drew last week, when it passed a DHS spending bill that would roll back Obama administration moves on immigration dating back to 2011.

A number of Republicans, led by Sen. Ted Cruz of Texas, want the GOP to take the most aggressive action possible, starting with the House bill. But Senate Republicans lack the votes to overcome a Democratic filibuster or an expected veto of the House plan — so the party’s top lawmakers hope to quickly settle on a fallback strategy.