Doing so could set off a chain of events on Capitol Hill that risks splitting the Republican conference, undercutting other parts of Trump's agenda and likely opening the administration's actions to legal challenges. It may also provide a clarifying moment that Republicans on the Hill have managed to avoid since Trump took office -- casting an up or down vote on whether to build the full-scale wall Trump desires.
According to federal law, Congress can rescind a presidential emergency declaration by passing a joint resolution. In the likely event that such a bill would be vetoed by the President, Congress could then override it with a two-thirds majority in the Senate and the House.
McConnell has been warning the White House publicly, and the President privately, that an emergency declaration is a bad idea. Not only is it opposed by Republican senators for ideological reasons, but it would also put those senators in a difficult political position of having to choose between supporting the President or what is seen as setting a dangerous precedent.
In the scenario Hill Republicans fear most, Trump declares a national emergency and House Democrats quickly pass a joint resolution against it, sending what amounts to a political grenade into the Senate. There, Republicans would be forced into either supporting a national emergency or rebuking a GOP president on his signature issue.
It would also force them to grapple with a handful of other thorny issues -- from executive overreach to separation of powers to raiding Pentagon money to pay for border security.
Faced with having to vote on a privileged resolution overturning Trump's emergency declaration for the wall, Cruz and Paul could either turn a blind eye to similar circumvention by Trump, inviting accusations of hypocrisy, or defy the President and risk looking weak on border security.
Many Capitol Hill Republicans would balk at assenting to a new norm on national emergencies, and worry about setting a precedent that future Democratic presidents could use to push a left-wing agenda item.
Two GOP sources cited Texas Sen. John Cornyn's public opposition to the move as a clear attempt to send that message to Trump.
"The whole idea that presidents -- whether it's President Trump, President Warren or President Sanders -- can declare an emergency and somehow usurp the separation of powers and get into the business of appropriating money for specific projects without Congress being involved, is a serious constitutional question," Cornyn told CNN on Feb. 4.
Constitutional issues aside, congressional appropriators in both parties guard their power closely, and an emergency declaration that redirected billions of taxpayer dollars would be seen as a clear theat. "Taken to an extreme, it would render the appropriations process meaningless," Republican Sen. Susan Collins of Maine told CNN.
And then there's the threat to swing-state GOP senators like Cory Gardner of Colorado and Martha McSally of Arizona, who risk alienating moderate voters if they have to vote to support Trump's national emergency, and inviting contempt from the President or conservative voters if they don't.
How would this nightmare play out? Declaring the emergency may prompt a House committee -- probably Judiciary, which oversees immigration issues -- to debate and vote on a resolution rescinding the emergency. After the Democratic majority there would recommend the resolution, the full House would vote and almost certainly pass it.
That would kick off 15-day timeline for the corresponding Senate committee to consider the resolution. The majority-Republican committee would make its recommendation and the resolution would become pending business on the Senate floor, requiring a vote within three days.
There are 53 Republican senators, so it would take just four GOP defections to pass the resolution out of the Senate. If it passes, Trump would still have to sign the resolution to rescind the emergency. He would almost certainly veto it. Congress could still override that veto, though that would take a two-thirds majority in both chambers. That's a much tougher task, requiring at least 55 Republicans in the House and 14 Republicans in the Senate to break with Trump.
At that point, though, Trump and plenty of members of his own party will have found themselves at odds with each other on the very issue Trump's has staked his presidency on. Avoiding such a scenario, then, would be a top political priority for Republican lawmakers. Otherwise, Hill Republicans will be stuck between a President determined to get his wall and newly empowered House Democrats determined to deny it to him.
CNN's Phil Mattingly and Dana Bash contributed to this story.