But the House measure, put forth by a group that includes two committee chairmen — Judiciary’s Robert W. Goodlatte of Virginia and Homeland Security’s Michael McCaul of Texas — was far more expansive.
It would require employers to use an Internet-based system, known as E-Verify, to confirm that they are hiring only legal workers; crack down on so-called sanctuary cities by denying them federal grants; allow for the detention of minors who are arrested at the border with their parents; and toughen sentences for criminals who have been deported and return illegally.
The measure would end the diversity visa lottery program, as Mr. Trump wants, and end family-based migration for all relatives other than spouses and minor children. It would offer three-year renewable work permits to DACA recipients, without offering them a path to citizenship.
The new House Republicans’ stand underscored the uncertainty about immigration. Mr. Trump’s positions vacillate daily. And members of both parties are divided. Some Democrats are pressing for confrontation, while others seem to fear a political backlash. Some Republicans are searching for compromise against a conservative tide of anti-immigrant fervor.
Lorella Praeli, the director of immigration policy and campaigns at the American Civil Liberties Union, described the House legislation as a “collection of hard-line provisions designed to sabotage, rather than advance, the possibility of a bipartisan breakthrough.”
It was not clear if the proposal would ever come up for a vote in the House, especially after Tuesday’s White House meeting established the parameters for a bipartisan deal. And it is all but certain to have no future in the Senate, where immigration legislation would need 60 votes for passage and therefore could not make it through the chamber with only Republican support.
The immigration debate on Capitol Hill grew more complicated after a federal judge ruled Tuesday that, for now, the Trump administration could not end the DACA program.
The judge, Judge William Alsup of Federal District Court in San Francisco, wrote that the administration must “maintain the DACA program on a nationwide basis” while lawsuits challenging the decision to end the program move ahead.
Mr. Trump lashed out on Twitter, saying that the United States court system was “broken and unfair,” and members of both parties and immigration activists struggled to understand the legal and political implications of the judge’s ruling.
Officials at the Department of Homeland Security, which administers the program for young immigrants, said they are preparing to follow the order even as they await possible action by the Justice Department to appeal the ruling in the days ahead.
Administration officials expressed confidence that the ruling would be overturned on appeal. And activists urged young immigrants not to take any actions to try to renew their DACA benefits until the legal picture is clearer.
Still, both parties insisted that the legal case is a distraction to the only real, permanent solution for the young immigrants brought to the United States as children: legislation passed by Congress and signed by the president.
Senator Jeff Flake, Republican of Arizona, said he was concerned that as a result of the ruling, “some will say, ‘Now we’ve got a lot of time.’”
Democrats tried to keep up the pressure for quick action.
“Let me be very clear: The ruling last night in no way diminishes the urgency of resolving the DACA issue,” said Senator Chuck Schumer of New York, the Democratic leader. He added that delay was “a tactic employed by those who do not wish to see a deal.”
Democrats are pushing for a deal on DACA by Jan. 19, when government funding is set to expire. The stakes are significant: If talks break down and lawmakers cannot, at a minimum, approve another stopgap spending bill, much of the government would shut down in less than two weeks.
Complicating matters, both parties face their own internal differences over how to proceed on immigration. Just as Republicans have to contend with more hard-line members, some liberal Democrats are adamant that they will not vote for a government funding measure without a resolution on immigration. Liberal activists have pushed Democrats to withhold their votes for government funding if a DACA deal remains elusive, in order to maximize their leverage.
But for Senate Democrats up for re-election this year in states that Mr. Trump won, a government shutdown could be politically perilous, especially if it is over immigration.
“You know what? Let’s think positive,” said Senator Jon Tester, Democrat of Montana, when asked about voting for a spending bill if DACA has not been resolved.
Senator Claire McCaskill, Democrat of Missouri, said, “I’m not big on drawing lines in the sand.”
Ms. McCaskill suggested the court ruling could actually help induce a deal more quickly. “I think everybody, especially the Republicans, would prefer us to decide this than the courts,” she said.
As they try to find a bipartisan solution on immigration, lawmakers have also had to try to decipher what exactly would be acceptable to Mr. Trump, given that he holds the veto pen.
During the 2016 presidential campaign, he vowed in a speech that August to “immediately terminate” DACA, adding that anyone who entered the United States illegally would be “subject to deportation.”
After he won the election that fall, he walked back that hard-line position and said, “We’re going to work something out.” Dreamers, he reiterated in April 2017, would be allowed to stay and “should rest easy.” On Sept. 5, Mr. Trump ended the program.
At times, the reversals occurred in the span of a day.
On Tuesday, in a meeting with a bipartisan group of lawmakers, Mr. Trump signaled a willingness to sign any deal worked out between Democrats and Republicans.
“I mean, I will be signing it. I’m not going to say, ‘Oh, gee, I want this or I want that,’” he said.
And on Wednesday, Mr. Trump gave seemingly contradictory stances in a matter of seconds. He affirmed that he would be open to signing “just about any immigration deal,” as a reporter put it, that lawmakers sent him. But asked if he would be willing to sign a deal that does not include funding for a border wall, he responded, “No, no, no, it’s got to include the wall.”