Washington (CNN) As President Donald Trump raises the possibility of invoking a national emergency to secure border wall funds, congressional Democrats are warning against such a move, while Republican lawmakers sound wary as some express concern over how that could play out or outright opposition.

"I think that would be a mistake," said Republican Sen. Roger Wicker of Mississippi Wednesday, of the possibility that the President might declare a national emergency.

"Members have communicated that to the President in a number of ways," Wicker said.

As the partial government shutdown -- triggered by a standoff between the President and congressional Democrats over a border wall -- drags on, the President has repeatedly floated the prospect that he may declare a national emergency in an effort to obtain the money he wants for a wall that Democrats refuse to give.

Prospects for any kind of a deal appeared even more unlikely on Wednesday after Trump declared that a meeting with congressional Democratic leaders Nancy Pelosi and Chuck Schumer to discuss the shutdown "a total waste of time." The President is asking for more than $5 billion in border wall funding, but Democrats have made clear they do not intend to meet that demand.

The President made a direct appeal to the American public on Tuesday evening in a prime-time address to the nation where he described the situation at the border as a "crisis." In a sign of how entrenched both sides remain, Pelosi countered the speech in remarks of her own along with Schumer, saying that Trump "must stop holding the American people hostage."

Trump kept the door open to declaring a national emergency on Wednesday, saying that he "may do that at some point" after leaving a meeting with Republican lawmakers on Capitol Hill.

But some Republican lawmakers have expressed concern over the possibility that the President might utilize military funds if were to go down that path. If Trump does declare a national emergency, he would have access to a pool of money inside the Department of Defense. Under Title 10 of the US Code, Trump could use funds "that have not been obligated" within the Defense Department's budget.

Republican Rep. Mac Thornberry, the top Republican on the House Armed Services Committee, made it clear Tuesday that he feels border security should be a responsibility of the Department of Homeland Security, not the military.

"I think border security is very important. It is not a responsibility of the Department of Defense," he said when asked specifically about using military construction money for the wall.

"I believe the Department of Homeland Security should be fully funded, including funding for physical barriers for people and technology, so that we can control who and what comes across our border," he added.

Other Republican lawmakers have pointed out that a national emergency declaration from the President would inevitably face legal challenge.

Sen. John Cornyn told CNN on Monday that he is "confident" the President could declare a national emergency, but said that he believes that would make the situation "more complicated," in part because of legal efforts to oppose it.

"What that may mean in terms of adding new elements to this -- in terms of court hearings and litigation that may carry this on for weeks and months and years -- to me injecting a new element in this just makes it more complicated," he said.

Asked if it would be wise for Trump to make an emergency declaration over the wall, Republican Sen. Rob Portman of Ohio said, "No. It's a much wiser idea to negotiate something out, because if it's done it will be tied up in the courts for a long time, and it would not be successful in achieving the objective. Plus it's cumbersome."

It's still unclear what route the administration would go if they choose to invoke emergency powers to attempt to start financing construction of portions of the wall. Amid that uncertainty, some Republicans are holding off on saying whether they would support such a move -- instead saying they will wait to see what happens before making a judgment.

Republican Sen. Ted Cruz of Texas said on Wednesday that he is planning to "wait and see" what the President does and won't "prejudge" the scenario when asked if the President has the legal authority to declare a state of emergency over the wall.

"I think it's critically important that we secure the border," Cruz said, but he added, "I'm also a constitutionalist. I have long argued that the president - any president Republican or Democrat -- should be bound by the Constitution and by federal statute and so I want to wait and see what the President does and what legal justifications he puts forth and assess them on their merits."

"I don't want to prejudge it until he actually acts," Cruz said.

Democrats, meanwhile, are sounding the alarm against the possibility of a national emergency declaration.

Democratic Sen. Dick Durbin of Illinois, a member of Senate Democratic leadership, met on Wednesday with acting defense secretary Patrick Shanahan "to warn him on how serious such a move would be, how legally dubious it would be, and to caution him that such an event will impact the Department of Defense's (DoD) overall relationship with Congress," the senator's office said in a statement.

When asked what Democrats would do if the President circumvents congress by invoking a national emergency over the wall, Durbin predicted that Trump would "face a challenge" during an interview on CBS' "Face the Nation" on Sunday.

"He's faced so many lawsuits when he ignores the law and ignores tradition and precedent and just goes forward without any concern. He'll face a challenge, I'm sure, if he's oversteps what the law requires when it comes to his responsibility as commander in chief."

Democratic Rep. Adam Schiff, the chair of the House Intelligence Committee, called the possibility a "non-starter" in an interview with CNN's Jake Tapper on "State of the Union" over the weekend.

"That's a non-starter and I think what the President needs to do more than anything else as he's painted himself into a corner on this thing is figure out how to un-paint himself into a corner. We need to re-open the government" Schiff said.