But there are questions about how broad, or narrow, the measure should be. Democrats want to shield young immigrants who were brought to the country illegally as children from deportation and offer them a path to citizenship, and could accept what they call “reasonable border security” provisions, such as electronic surveillance, including drones and drug detection equipment, and rebuilding roads across the border.
Mr. Trump and Republicans would like to expand the measure so that it sharply limits what he calls “chain migration,” the longstanding policy that allows one family member to sponsor another to come to the United States, and ends the so-called visa lottery, which aims to diversify the immigrant population by selecting applicants randomly from countries with low rates of immigration. Mr. Trump called the lottery “a disaster.”
Mr. Trump repeated those demands in a tweet Thursday night.
“We must BUILD THE WALL, stop illegal immigration, end chain migration & cancel the visa lottery,” he said. “The current system is unsafe & unfair to the great people of our country — time for change!”
The president also used Thursday’s meeting to reiterate his demand for a border wall.
“We need a physical border wall,” he said. “We’re going to have a wall — remember that — we’re going to have a wall to keep out deadly drug dealers, dangerous traffickers and violent criminal cartels. Mexico is having a tremendous problem with crime, and we want to keep it out of our country.”
Still, there appears to be some confusion about what, precisely, the president wants. After Thursday’s meeting, Senator James Lankford, Republican of Oklahoma, said Mr. Trump was not insisting on a massive physical wall.
“People want to paint his definition, that it’s some 2,000-mile-long, 30-foot-high wall of concrete,” Mr. Lankford said. “That’s not what he means, and that’s not what he’s trying to say.”
The senator continued: “There’s going to be border fencing in some areas, there’s going to be vehicular barricades, there’s going to be technology, greater manpower in some areas. Different parts, whether they’re mountainous or whether they’re open deserts, need different solutions.”
Senator Charles E. Grassley, Republican of Iowa, who also attended Thursday’s meeting, said Wednesday that he expected a “very good compromise” to be ready by the end of the month, and that it would have a mix of border security provisions.
“I hate to use the word ‘wall,’ because that implies you might want a steel wall for 2,000 miles,” said Mr. Grassley, who is chairman of the Judiciary Committee, which has jurisdiction over immigration. “We want border security. Sometimes it’s wall, sometimes it’s more border personnel. I t might be electronic surveillance, drones. The idea is that people come to this country with papers.”
An estimated 780,000 immigrants were covered under the Obama-era program, known as Deferred Action for Childhood Arrivals, or DACA. Participants were required to register with the government and pass background checks, in exchange for two-year renewable permits that allowed them to work or attend school.
Immigration advocates say 14,000 of them have already lost their protected status because they have been unable to renew permits issued more than two years ago.
“This is a crisis created by Donald Trump himself,” said Greisa Martinez Rosas, 29, who has a permit under the program and now works for United We Dream, an advocacy group. “He is now using my life and the lives of immigrants of United We Dream as ransom to move forward his anti-immigrant agenda.”
On Capitol Hill, five senators — three Republicans and two Democrats — have been meeting nearly every day for the past two months in an attempt to come up with an immigration agreement. Senator Richard J. Durbin, the No. 2 Democrat in the Senate, who is leading the talks, was tight-lipped about the group’s progress, saying only, “We’re continuing to meet.”
But one person familiar with the discussions said the talks could not conclude until Mr. Trump made his wishes clear. Mr. Lankford said he hoped Mr. Trump would put a proposal in writing before next week’s bipartisan meeting at the White House.
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