Top Republicans hinted Tuesday that they’ll try to bring up the House-passed DHS funding bill again after it failed to advance on a 51-48 vote, but they offered little clarity on how the new GOP-led Congress will ultimately produce legislation that doesn’t prompt a veto threat from the White House.

“I think we’ll give them an opportunity to vote on that more than one time,” John Thune of South Dakota, the third-ranking Senate Republican, said before the vote. “Just using the procedural rules to keep us from even debating it I think is just – I mean it’s just a disservice to people who care deeply about this issue on both sides.”…

GOP senators have said little publicly on what their fallback strategy would be when the House-passed DHS funding bill met an inevitable death in the Senate. The possibilities that have been mulled include short-term funding measures, a lawsuit challenging Obama’s actions and the passage of border-security legislation — although that last option has been panned by some conservatives who suspect that doing so would be a gateway to broader immigration legislation.