Inside a private luncheon with other Republican senators after the Pence gathering, Tillis made the case that if the White House agreed to some form of Lee's legislation, that would mean a president would be giving up some of his executive powers - a rarity, according to a person briefed on the closed-door discussion.
If Trump's border emergency stands, he could divert $3.6 billion from military construction projects to build border barriers, even though Congress had voted to limit him to less than $1.4 billion for such construction. That would be enough to kill the resolution in the Senate, provided no other Republican senators oppose of Trump's declaration or alter their position.
Still, it would call attention to congressional opposition to one of his chief campaign pledges: building the wall.
Congress does not have the votes to override Trump's veto.
The Senate is require to vote on the resolution by Friday.
Canada grounds Boeing 737 MAX jets following Ethiopia crash
Ameen Ismail Noormohamed (right), who was killed during the Ethiopian Airlines crash on March 10, poses with his son Naheed. All 157 people aboard the Boeing 737 MAX 8 died when it crashed in Addis Ababa on Sunday.
Lee had introduced a second measure that would end future emergency declarations after 30 days unless Congress votes to extend them.
Sen. Josh Hawley (R-MO), a constitutional lawyer and a freshman senator, came out in favor of President Trump's move to secure the border through the National Emergencies Act.
In a statement issued later, the Utah senator said, "Unfortunately, it appears the bill does not have an immediate path forward, so I will be voting to terminate the latest emergency declaration".
"We're going to have a news conference at the border over the next three weeks", Trump told reporters at a briefing at the White House on drug trafficking. "Let's tell the president he cannot use his overreaching power to declare an emergency when he couldn't get Congress to do what he wanted".
"Republican Senators are proposing new legislation to allow the President to violate the Constitution just this once in order to give themselves cover", she said. If Trump would commit to signing a bill that would handcuff future emergency declarations, some of those senators might support his border declaration.
Brown said he doesn't object to finding a "long term answer", to keep presidents from declaring emergencies in non-emergency situations, but said he didn't know of any cases where past presidents abused the law. The White House, so far, has declined to provide that commitment.
A vote on Lee's plan was expected after Congress returns from a recess later this month. Although he said Trump has the power for his current declaration, McConnell said Republican senators were now questioning: "Was it too broad back in the '70s when it was passed?"