President Trump said Tuesday that he would “love to see a shutdown” over immigration if Democrats don’t come to the table to negotiate on border security and other stiff enforcement measures, escalating the stakes with less than four weeks to go before a deadline for action.

“If we have to shut it down because the Democrats don’t want safety … let’s shut it down,” the president said at a White House roundtable discussion on MS-13 gang violence and illegal immigration.

The tough talk seemed to upend congressional Republicans’ carefully crafted strategy of avoiding shutdown threats, hoping to pin blame on Democrats if the government did tumble into another stoppage.

Rep. Barbara Comstock, a Virginia Republican who represents a number of federal employees, told Mr. Trump that she thought lawmakers could reach an immigration deal without resorting to the worst-case scenario.

“We don’t need a government shutdown on this,” she said.

Mr. Trump was insistent, however: “You can say what you want. We are not getting support of the Democrats.”

The White House said Mr. Trump isn’t cheerleading for a shutdown and hopes the government keeps running, contingent on a Democratic agreement to a two-year budget and to laws cracking down on illegal immigration as part of a deal to legalize immigrant Dreamers.

It’s not clear what shutdown Mr. Trump envisions.

A government funding deadline is looming this week, but Democrats agreed to decouple that from the immigration talks.

Meanwhile, the deadline Mr. Trump set for dealing with Dreamers is March 5, which is when his phaseout of the Obama-era Deferred Action for Childhood Arrivals deportation amnesty fully kicks in. But no government funding or debt deadline is tied to March 5.

“We are not advocating for the shutdown,” said White House press secretary Sarah Huckabee Sanders. “The president’s encouraging people to do their jobs.”

Senate Minority Leader Charles E. Schumer, New York Democrat, said Mr. Trump’s own words belie that.

“We had a Trump shutdown. Nobody wants another, maybe except him,” Mr. Schumer said.

The Senate is slated to begin an immigration debate over the next week — though it’s unclear what a bill would look like.

On Tuesday, Majority Leader Mitch McConnell, Kentucky Republican, wasn’t certain what bill he would even bring to the floor to use as a basis for the debate.

“There’s no secret plan here to try to push this in any direction,” Mr. McConnell said.

He later added, “We just don’t know where 60 votes are for any particular proposal, and so we’re going to find out.”

House Speaker Paul D. Ryan, Wisconsin Republican, said he won’t take up a bill that doesn’t have Mr. Trump’s backing.

Republicans said they have been waiting for Democrats to make a counteroffer to Mr. Trump’s four-prong plan to enact a generous amnesty for 1.8 million illegal immigrant Dreamers in exchange for major policy changes including a border wall and more authority to speed deportations, an end to the visa lottery and limits to the chain of family migration.

Democrats say Mr. Trump’s plan is a nonstarter. They want a “clean” vote on granting citizenship to as many as 3 million illegal immigrants.

The shutdown talk overshadowed Mr. Trump’s efforts to draw attention to the threat of MS-13, a gang that acting Assistant Attorney General John Cronan said is stocked by illegal immigrant “savages.”

“MS-13 can simply replenish its jail population by sending more and more members across our borders,” Mr. Cronan said.

The administration argues that the pipeline relies heavily on the flow of illegal immigrant minors who have surged across the border over the past five years, taxing the ability of the Homeland Security Department to deal with them.

“When they come to our border, I have to let them in,” Homeland Security Secretary Kirstjen Nielsen said at the White House roundtable.

“This is unique to our country, and it’s got to change,” Mr. Trump said.

Immigrant rights advocates don’t defend the gangs but say Mr. Trump is giving immigrants an unfair rap by pointing to MS-13 as part of the debate.

The Democratic National Committee called Mr. Trump’s remarks an attempt to “fearmonger,” saying he was “falsely [conflating] MS-13 gang members with undocumented immigrants.”

Data from a major anti-gang operation last year, though, suggested a link.

A series of raids dubbed Operation Raging Bull netted 214 MS-13 gang members in California. Of those arrested, 193 were illegal immigrants and 60 had entered the country as unaccompanied minors, according to U.S. Immigration and Customs Enforcement.

American citizens accounted for just 16 of the gang members, and five were foreign nationals with legal residency in the U.S.

Angel Melendez, special agent in charge in Homeland Security Investigations’ New York office, said a stunning 30 percent of MS-13 members arrested in recent anti-gang operations came to the U.S. as part of the surge of unaccompanied alien children that overwhelmed the Obama administration in its final years in office.

Of 40,810 unaccompanied alien children who arrived in the U.S. and were granted initial entry in 2017, more than half were males ages 13 to 17 from the Central American countries that serve as the source for MS-13, Mr. Melendez said.

He said not all of unaccompanied children are gang members, but MS-13 is “looking at these 20k AUC that came into the United States as potential recruits to refill their ranks.”

• Dave Boyer and S.A. Miller contributed to this article.

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