The White House said Thursday that it wants to see money for President Trump’s border wall included in the spending bill Congress must pass next week — a demand Democrats said sours negotiations and makes a government shutdown more likely.

The demands mark a reversal for the administration, which had been saying it found enough money to build prototypes this year and wouldn’t need a major infusion of cash until next year.

But White House Budget Director Mick Mulvaney said in an interview with The Associated Press on Wednesday that the wall and the money for more immigration agents are priorities.

“We know there are a lot of people on the Hill, especially in the Democratic Party, who don’t like the wall, but they lost the election. And the president should, I think, at least have the opportunity to fund one of his highest priorities in the first funding bill under his administration,” Mr. Mulvaney said.

The White House issued its demand just days after Democrats insisted that the spending bill include billions of dollars to prop up Obamacare. Democratic aides signaled that they wouldn’t accept a bill without the cost-sharing payments intended to keep insurers invested in the health care law.

With Mr. Mulvaney’s demand, both sides now appear to be entrenching.

“Everything had been moving smoothly until the administration moved in with a heavy hand,” said Matt House, a spokesman for Senate Minority Leader Charles E. Schumer, New York Democrat. “Not only are Democrats opposed to the wall, there is significant Republican opposition as well.”

Congress is once again on the clock, racing an April 28 deadline for passing a funding bill to last the rest of the fiscal year, which ends Sept. 30. Without a bill, the government would face a partial shutdown ending nonessential services.

The Trump White House will begin issuing plans to agencies Friday in preparation for a shutdown.

It would be the second shutdown in four years. In 2013, congressional Republicans orchestrated a shutdown to try to stop funding of Obamacare. That 16-day shutdown ended after Republicans conceded.

Democrats are convinced that Republicans would be blamed for a shutdown this time as well. As the minority party, though, Democrats would play the bigger role in mounting an obstruction.

Mr. Trump, at a press conference Thursday, said he is not angling for a shutdown.

“As far as keeping the government open, I think we want to keep the government open. Don’t you agree?” he told reporters.

Mr. Trump promised a border wall during the presidential election campaign and asserted that Mexico would pay for it. He now acknowledges that U.S. taxpayers will foot the bill upfront.

Some lawmakers in Washington have questioned whether a wall is needed. During the first two months of Mr. Trump’s administration, there has been an astounding drop in the flow of people attempting to jump the southwestern border.

Trump administration officials take credit for the drop, saying the president’s get-tough rhetoric has changed the calculations of would-be migrants from Latin America, but they insist the wall is needed to make sure those gains are maintained.

“It’s absolutely essential that we build a wall,” Homeland Security Secretary John F. Kelly said as he toured the border in El Paso, Texas, on Thursday with Attorney General Jeff Sessions.

Mr. Kelly said a wall must be combined with technology and agents to back it up.

His department has found $20 million in funding that it will use to build wall prototypes. Bids were due earlier this month, and officials say testing will take place this summer in San Diego.

The prototypes have to be at least 18 feet high, though 30 feet is the preferred height, and should be able to withstand breaching attempts for up to four hours, according to the contracting documents. The wall also must look imposing, the documents said.

Senate Democrats this week released a report saying the wall could stretch 1,827 miles, or most of the way across the 1,950-mile border, at a cost of nearly $70 billion. Homeland Security officials said until they decide on a final design and locations, they can’t begin to make an estimate — though they said the Democrats’ figures were baseless extrapolations.

One immigration analyst said the administration had been taking a low-key approach to wall funding until early this week, when it suddenly became a priority in negotiations.

It could be a bargaining chip designed to counter Democrats, who just last week insisted that the Obamacare money be included.

Mr. Mulvaney said neither he nor Mr. Trump is enthused about the Obamacare money but that they could accept it if Democrats are willing to deal on Republican priorities.

“If it’s important enough to the Democrats, we’d be happy to talk to them about including that in sort of some type of compromise,” the budget director told AP.

Mr. Trump’s immigration plans have been among the most controversial issues of his young administration.

Immigrant rights activists this week complained that his agents deported an illegal immigrant Dreamer, 23-year-old Juan Manuel Montes, who was supposed to be protected under Mr. Obama’s 2012 deportation amnesty.

Mr. Kelly, speaking at the border, said Mr. Montes was deported back to Mexico after breaking terms of the amnesty, known officially as Deferred Action for Childhood Arrivals, or DACA.

“There was a time in his life that this individual was a DACA resident, but he gave that up by his behavior and his illegal actions. He’s no longer covered by the DACA arrangement,” Mr. Kelly said.

The secretary didn’t specify those illegal actions, but his department this week said Mr. Montes had amassed several convictions for driving without a license and one conviction for shoplifting. In addition, Mr. Montes left the U.S. without permission, which was a violation of DACA.

The young man’s defenders, though, said the Homeland Security Department is fabricating its case.

“Right now, Trump agents are lying about Juan’s story. They are smearing his name,” said Greisa Martinez Rosa, advocacy director for United We Dream, a group that defends DACA recipients.

She compared Mr. Montes’s deportation to race-related police slayings. “What they are doing to Juan is the same thing abusive law enforcement do to people of color who are killed,” she said.

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