Currently, Congress has to pass a resolution of disapproval to block Trump's emergency declaration on constructing the U.S.-Mexico border wall. Trump is expected to veto the resolution and neither chamber is expected to be able to override that veto.

Trump's emergency declaration has sparked a widespread discussion among Senate Republicans about if they should change the National Emergencies Act. Though Republicans largely support Trump on border security, they are concerned that a future Democratic president will use his precedent to force through action on issues like climate change or gun control.

Lee's legislation would not impact Trump's current emergency declaration on the wall but, if passed, would impact any future emergency declarations.

Senators are discussing changes to the emergency powers law with the White House but it's unclear if Trump would support reining in his own executive authority, and he's consistently bristled at attempts by Congress to force his hand on legislation.

Blunt, a member of GOP leadership, said Trump had raised concerns about executive overreach by previous presidents and that supporting Lee's legislation would give him a chance to get back in line with his campaign rhetoric.

"Well, the president had problems as a candidate with the Obama overreach, so he's been on record for some time on this topic," he said. "I think this will give him a chance to get back to where he was three years ago."

"We're looking at some ways to revisit the law. There's a lot of discomfort with the law. … Was it too broad back in the '70s when it was passed? So yeah, we're discussing altering that," McConnell told reporters during a weekly press conference.

Asked if he would support Lee's legislation, McConnell added that he "may well."