A national emergency declaration by President Donald Trump over border security could wind up hurting Republicans, GOP Sen. Marco Rubio told CNBC on Wednesday.

The Florida Republican contended that Trump was elected on the promise of building a wall along the U.S.-Mexico border and the president has to "keep that promise." But "we have to be careful about endorsing broad uses of executive power," he added. "I'm not prepared to endorse that right now."

Such a declaration would set a precedent, Rubio said. "If today, the national emergency is border security ... tomorrow the national emergency might be climate change." Declaring a national emergency, which would certainly face legal hurdles, could give Trump the ability to use the military to build the wall instead of getting Congress to approve the funding for it.

Rubio joined "Squawk Box" as the partial government shutdown entered its 19th day due to the stalemate between Trump and Democrats on funding for a border wall. In a prime-time TV address Tuesday night, Trump made his case for a wall, highlighting multiple grisly examples of Americans allegedly murdered by undocumented immigrants.

Rubio said he would have liked the president to clarify during his address that the request for $5.7 billion in border security funding is not for a wall or a loan. "It's $5 billion to fund the top 10 projects on a border security program by the experts and the people in charge of doing that," Rubio said.

Asked whether he supports Trump's position, which includes the wall, Rubio said, "Anything that makes the border more secure, I'll be in favor of supporting."

Ahead of Trump's address, Vice President Mike Pence met privately with House Republicans, urging them to "stand strong" and insisting the White House wants to negotiate, people familiar with the conversation told The Associated Press. The White House said Trump and Pence will meet with Republican senators at the Capitol on Wednesday.

Senate Majority Leader Mitch McConnell said Trump's proposal to increase border security through physical barriers "suits the reality on the ground" along the U.S.-Mexico border.

Democrats doubt a wall would do much to solve the issue of illegal immigration. Democratic leaders Sen. Chuck Schumer and House Speaker Nancy Pelosi said a wall would be ineffective and unnecessary.

"The president has chosen fear. We want to start with the facts," Pelosi said, arguing Trump is rejecting bipartisan proposals to end the ongoing partial government shutdown over his "obsession with forcing American taxpayers" to pay for "a wall he always promised Mexico would pay for." Schumer added, "Separate the shutdown from the arguments over border security."

Rubio, the son of Cuban immigrants, sought the GOP presidential nomination in 2016 but ultimately dropped out after losing the Florida primary to Trump.