President Trump edged closer to declaring a national emergency Thursday that would allow him to build a barrier on the southern border without Democrats’ approval, even suggesting he might outsource some of the work to the state of Texas.

With no negotiations set for resolving the 20-day-old partial government shutdown, Mr. Trump traveled to a Border Patrol station in McAllen, Texas, to meet with agents and highlight the immigration and trafficking crisis. He said the collapse of talks with Democratic leaders a day earlier has nearly persuaded him of the need to declare an emergency.

“I have the absolute right to declare a national emergency,” the president said. “If this doesn’t work out, probably I will do it. I would almost say definitely. I’m prepared for anything.”

At the Capitol, Vice President Mike Pence told reporters that the president “is going to get this done one way or the other.”

“There’s no wall, there’s no deal,” the vice president said.

An attempt by a small group of GOP senators to strike an agreement trading wall funding for full legalization of “Dreamers” here under the Obama-era DACA program fell apart Thursday afternoon.

Democrats continued to push for an immediate reopening of the federal government, saying they won’t negotiate until nine shuttered departments are back in business.

House Democrats passed two bills Thursday to reopen some of those departments, but Senate Majority Leader Mitch McConnell blocked them in his chamber, saying Mr. Trump would have vetoed them anyway.

House Speaker Nancy Pelosi accused Mr. Trump of creating a “distraction” from his “other problems” — an apparent reference to special counsel Robert Mueller’s investigation.

“I don’t even know if the president wants the wall. He wants a debate on the wall,” she said.

For his part, Mr. Trump flew about 1,700 miles Thursday to the Mexican border to show that he wants a wall.

Following up on his address to the nation about a border crisis, the president met with Border Patrol agents and other law-enforcement officials in Texas.

During a roundtable discussion, Mr. Trump noted that Texas Lt. Gov. Dan Patrick had suggested providing the state with “a relatively small amount of money, and they’ll build a wall themselves.”

“I thought that was not the worst idea I’ve ever heard,” he told Mr. Patrick. “Although I still think I can do it cheaper than you. I do like the idea. We’re going to look at a couple of ways of doing it, where you guys get [the wall] up. I like that idea. We’ll take a look.”

The president also heard from relatives of two law-enforcement officers killed by illegal immigrants — police Cpl. Ronil Singh of Newman, California, and Border Patrol Agent Javier Vega Jr. of Texas.

Marie Vega recounted how her son was shot and killed during a robbery while on a family fishing outing in 2014.

“A parent should not have to bury their child,” she told the president. “We need the wall. We need tougher judges.”

Reggie Singh spoke of his brother’s 5-month-old son having to grow up without a father after the corporal was shot and killed on the day after Christmas, allegedly by an illegal immigrant he pulled over in a traffic stop.

“What my family is going through right now, I do not want any other law-enforcement family to go through,” Mr. Singh told the president.

Others joining the president were Sens. John Cornyn and Ted Cruz, Texas Republicans; Homeland Security Secretary Kirstjen Nielsen; Customs and Border Protection Commissioner Kevin McAleenan; and the head of the U.S. Army Corps of Engineers, Lt. Gen. Todd T. Semonite.

The Army Corps would likely be in charge of wall building should Mr. Trump claim emergency powers, and the White House directed them to examine supplemental funding to see what money could be used if Mr. Trump declares a national emergency.

And the president pushed back on critics who say he’d be circumventing Congress, saying Capitol Hill approved the National Emergencies Act that would be the basis for his claims.

He said declaring an emergency would be “the easy route for me,” and said he hasn’t done it because he would still like to get a deal with Congress.

The president blamed Democrats for the partial government shutdown, rebutting their criticism that he has “manufactured” a crisis on the border to fulfill his 2016 campaign pledge to build a wall. Mr. Trump reiterated that their opposition is geared toward his re-election bid in 2020.

“The Democrats don’t care about crime. They’ve been taken over by a group of young people who, frankly, in some cases, I’ve been watching, I actually think they’re crazy. They have been taken over by a group of people that don’t care about gangs, they don’t care about human trafficking and drugs, they don’t care about anything. I tell you what, they have gone crazy,” he said.

The White House says it’s made two different offers to Democrats, one in the early days of the shutdown and another after last weekend’s work at Camp David.

Mr. Pence said the first was rejected, and there’s been no movement on the second.

That latest deal calls for combining $5.7 billion in border wall money with more immigration detention beds and more Border Patrol agents, reviving a program to allow children to apply for asylum from Central America, and spending more money on medical care for migrants in U.S. custody and for screening traffic coming through official border crossings.

Those last three are Democratic wish-list items.

“It’s hard for the American people to accept a party that won’t even negotiate an agreement over border security is committed to border security,” Mr. Pence said.

Democrats say they’re willing to talk, but won’t actually negotiate as long as part of the government is shut down.

Mrs. Pelosi also said Wednesday’s White House meeting, where she refused any talk of wall money and Mr. Trump walked out, was staged.

“I think it was a set-up so he could walk out,” Mrs. Pelosi told reporters.

Opponents say Mr. Trump’s possible declaration of a national emergency would be met with an immediate court challenge to block his authority to build a wall unilaterally. He said the White House counsel’s office has advised him that such a move is “100 percent” legal.

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