That could complicate efforts to pass a permanent solution for the Deferred Action for Childhood Arrivals policy that the President is ending, after Democrats have signaled they could cause trouble on government funding if no progress is made by the time it runs out in December.
"We definitely ruled out putting any kind of DACA package on the omnibus bill. Period -- end of story," Arkansas GOP Sen. Tom Cotton told reporters afterward in the Capitol.
"This is not going to be part of the year-end omnibus or CR and it's (going to be) coupled with some other border security and interior enforcement," Senate Majority Whip John Cornyn added.
Also at the White House meeting were Senate Judiciary Chairman Chuck Grassley and immigration bill authors Sens. Tom Cotton, David Perdue, James Lankford, Thom Tillis and Lindsey Graham, who is leading negotiations with Senate Minority Whip Dick Durbin.
Later Thursday, Trump did not address DACA but repeated his call for Congress to end the diversity visa lottery and so-called "chain migration," or extended family sponsorship.
"The people in that lottery are not that country's finest," Trump said at the White House. "Congress must end chain migration. Ultimately we want a system that's merit based."
A source familiar with the morning meeting said it signaled progress but no major breakthroughs, and while the diversity lottery came up and the President would prefer a permanent legislative fix, the White House issued no red line on how it would get done.
Asked to react to the statements by Cotton after the meeting on Thursday, Senate Minority Leader Chuck Schumer shrugged.
"We're going to do everything we can for DACA," he said.
Talks have so far produced no clear outlines of a deal to protect the young undocumented immigrants brought to the US as children, who will begin losing their protections in March under Trump's sunsetting of the program.
The door was opened to some kind of end-of-year deal last week
, when House Speaker Paul Ryan raised the possibility in a closed-door meeting with a group of House conservatives, according to a senior GOP aide. A Ryan spokeswoman clarified it was not a set plan, and Ryan said in an interview later that day the timing was "open to debate."
With deep divisions in the GOP over how to approach such a deal, which all Republicans and Trump insist must include some type of border security and immigration enforcement with it, Democratic votes are likely required to pass something, and a spending deal would offer especially House Republicans some cover in voting on a proposal. It would also avoid a lengthy amendment process in the Senate.
Democrats have also signaled that they would consider boycotting a government funding deal in December unless DACA had been addressed, setting a potential shutdown showdown.
California Sen. Kamala Harris, a progressive Democrat increasingly talked about as a possible 2020 candidate, became the first Democratic senator last week to join Congressional Hispanic Caucus members who have discussed refusing to vote for funding with DACA unresolved.
There may still be Republican support for a spending deal, if necessary. Arizona Republican Sen. Jeff Flake, a longtime proponent and author of immigration reform, said the party should keep its options open when asked about the group's conclusion.
"I don't know if we ought to draw that line. That deadline's coming up in early March, I say the sooner the better. I don't know why we'd wait."
Diversity lottery and other sticking points
Lawmakers shrugged off the idea that a DACA deal will be complicated by Tuesday's New York terrorist attack, which was allegedly committed by a 29-year-old Uzbek national who came to the US on a diversity lottery visa.
Trump has endorsed a proposal from Cotton and Perdue that would do away with that program, which doles out 50,000 of about 1 million green cards to the US given annually to countries with low levels of US migration, would sharply cut family-based green cards and overhaul the existing employment-based green card system.
"I think the President wants both of those included," Cornyn said after the meeting, but didn't say it was a necessity. "We'll see what the Democrats are willing to negotiate. That's what this is, the beginning of a negotiation."
Cotton, Lankford and Cornyn all told reporters that the bill needed to be narrow enough to avoid getting bogged down with every immigration proposal, and Cotton said the diversity lottery didn't "necessarily" need to be included.
"I wouldn't say we ruled anything in or out there (on policy)," Cotton said.
Cornyn said further negotiations come next, predicting a deal in January or February.
"Now the work of getting together with our House colleagues is the next step, and then we'll engage Sen. Durbin and the Democrats to see if we can work something out," Cornyn said.