Late Friday, Congress passed a one-week stopgap bill in an effort to avert the shutdown, with leaders in both parties pushing back against House conservatives who were angry that the budget measure left Obama's plan alone.
The temporary funding measure was passed by a bipartisan vote of 357-60 in the House just a little past an hour after the also-Republican-controlled Senate approved the Homeland Security funding measure without even the need for a roll call vote.
Meanwhile, Senate Republicans say it's time to move on from the battle over Obama's executive action, after a federal court ruling in Texas that declared it violates the Administrative Procedure Act. The Obama administration is fighting the ruling, but Senate Republicans believe the Fifth Circuit Court of Appeals will uphold the lower court's decision.
"Why don’t we just look at the court decision in Texas, declare victory and move on?" said Arizona GOP Sen. Jeff Flake.
And Sen. Lindsey Graham, R-S.C., agreed that "when the judge ruled, that was the way I wanted to end it."
But House conservatives are holding out for a bicameral conference to allow the House and Senate to decide on a compromise that would keep funding Homeland Security while repealing some of Obama's executive order.
However, Senate Democratic Leader Harry Reid ruled that out on Thursday, and many Senate Republicans call those hopes misguided.
McConnell, R-Ky., and House Speaker John Boehner, R-Ohio, have not been in close communication, The Hill reports, with Boehner saying on Wednesday that he and McConnell have not spoken in two weeks.
Senate Republicans who are up against tough reelection battles in 2016 have also been pressing the House to accept the clean funding bill, which would extend DHS funding through September.
"I think the decisive arena is the courts," he said. "I think this is about holding our own ground and it's also a message to the Senate."
That message is that "you really do have to vote on things and you really do have to go to conference," Cole said. "You’re not going to just drop something on us the last day and run out of town."
Arizona Republican Sen. John McCain