Washington (CNN) President Donald Trump dug in Wednesday on his demand for border wall funding and sought to project a unified Republican front as he took to Capitol Hill to shore up the GOP caucus.

In a pair of public appearances at the White House and on Capitol Hill, Trump stressed the need for southern border wall to stem the flow of illegal immigration and argued it would be politically "foolish" for him to agree to reopen the government unless his demands are met.

Emerging from lunch with the Senate Republican caucus, the President insisted Republicans "are totally unified" behind his hardline strategy, even as a handful of Republican lawmakers made clear they have begun to question his legislative strategy.

While Trump insisted there were no dissenting views, Alaska Sen. Lisa Murkowski -- who on Tuesday called for reopening the government without border wall funding -- said she told Trump "people are starting to feel those consequences" of the shutdown.

Both privately and publicly, the President maintained his belief that he has the more potent argument.

"He urged us to remain unified," Murkowski said of Trump's remarks during the Wednesday lunch.

Even as he signed a legislation aimed at combating human trafficking, Trump argued that a wall is the only remedy.

"They say it's a medieval solution, a wall. It's true, because it worked then and it works even better now," Trump said during a bill signing event in the Oval Office.

But politics, too, were front of mind. He made clear that he is sticking by that demand in part because he could face a political backlash from some Republicans if he backs down.

"Right now, if I did something that was foolish like gave up on border security, the first ones that would hit me are my senators. They'd be angry at me," Trump said, expressing publicly what he's told lawmakers and allies over the past several weeks. "The second ones would be the House and the third ones would be, frankly, my base and a lot of Republicans out there and a lot of Democrats that want to see border security."

Trump's comments came shortly before he headed to Capitol Hill to shore up support among Senate Republicans during their weekly lunch, at a moment when some Republicans have begun to signal that Trump should sign appropriations bills that will reopen the government now rather than remain defiant on his border wall demands.

"The Republicans are unified," Trump said after the meeting with Senate Republicans. Trump also claimed it's Democrats who are more likely to cave.

So far, though, no Democratic lawmakers have called for their leaders to agree to fund the border wall to reopen the government.

The ongoing stalemate -- which entered its 19th day on Wednesday -- has led Trump to keep the door open to declaring a national emergency in order to appropriate existing government funds to build a border wall. Trump considered but ultimately decided against declaring the national emergency in his Tuesday prime-time address , but made clear on Wednesday he could still go that route.

"I really believe the Democrats and the Republicans are working together. I think something will happen, I hope," he said. "Otherwise we'll go about it in a different manner."

He added that he believes he has "the absolute authority to do a national emergency if I want" and said he would follow through "if I can't make a deal with people that are unreasonable."

The President also signaled an openness to a broader immigration deal and said the shutdown could be a "blessing in disguise" that would lead to a broader immigration deal.

"I would love to see a big immigration bill that would really take care of this situation," Trump said. "Right now, we have a problem and we have to take care of this. But we would like to see real immigration reform in this country."

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Despite those comments, the White House has yet to signal a willingness to reaching a broader deal that would include both funding for the wall and protections for undocumented immigrants, like Deferred Action for Childhood Arrivals recipients. The White House previously walked away from such a deal last year.

The White House is continuing a full-court press to argue that there is a "crisis" on the southern border and that a wall is the solution to that problem.

Trump and Democrats remain at an impasse over border wall funding, with Trump so far refusing to agree to a spending bill that does not include border wall funding and Democrats refusing to support legislation that does fund a border wall.

After meetings with lawmakers on Capitol Hill and at the White House on Wednesday, Trump will head to the US-Mexico border on Thursday to continue making his pitch.

But privately, Trump has said his photo op on the border will also likely do little to upend the stalemate.