But the meeting was boycotted
by one Republican who is actively negotiating with Democrats, Sen. Jeff Flake of Arizona, for not being bipartisan, and even the GOP lawmakers in the room did not all agree on how to hammer out remaining sticking points.
President Donald Trump called for a bipartisan meeting next week to follow, lawmakers said afterward, and Vice President Mike Pence personally called to invite Flake, who accepted.
Democrats, meanwhile, are keeping their options open -- doubling down on bipartisan negotiations and declining opportunities to draw red lines around some of the proposals.
The shape of a deal
Republicans who were in the meeting, including Sens. John Cornyn of Texas, Thom Tillis of North Carolina and James Lankford of Oklahoma, all described a similar set of ingredients. A deal should include a resolution for DACA -- which currently would be a path to citizenship for qualifying young undocumented immigrants, negotiators say -- along with beefed up border security that would include physical barriers, some limits to family-based visa categories and the end of the diversity visa lottery.
But there was disagreement over what all that consists of specifically.
South Carolina Sen. Lindsey Graham, who was at the White House meeting, and Flake -- who have been negotiating intensely with Democratic Whip Dick Durbin of Illinois and Colorado Sens. Michael Bennet, a Democrat, and Cory Gardner, a Republican -- both said Thursday that the "chain migration," or family-based migration, piece would be limited.
"We're not going to fix it all," Graham told radio host Hugh Hewitt on Thursday. "But the first round, there will be a down payment on breaking chain migration."
Flake told reporters that the negotiations were settling on limiting the issue of "chain migration" to the DACA-eligible immigrants protected in the eventual deal.
But Lankford flatly rejected that approach.
"No," he said when asked about Graham's characterization of talks. "This has to be broader than that, because if you're going to deal with chain migration, you deal with chain migration. ... I can't count on the fact that we're going to do another (bill) in six months to resolve the rest of it."
Lawmakers are discussing ending the diversity visa lottery but not erasing the 50,000 visas for legal permanent residency distributed through it annually. Graham said the deal would "use them more rationally" and Flake said it would be part of a trade for resolving a type of immigration protection for nationals of countries who suffer major disasters, which the Trump administration has moved to curtail.
And the border security piece still remained elusive, even as Trump continues to demand his wall. Lankford and Tillis made efforts to tell reporters that the "wall" piece does not mean a solid structure all the way across the entire southern border.
"That's not what he means. That's not what he's tried to say -- I think that's what people are portraying it as," Lankford said. But neither could describe what Republicans actually want out of a border deal, and they said they were still waiting for the White House to provide clarity on what it could and could not live with.
"What we did today that I thought was truly (a) breakthrough ... we saw the President assume leadership on this issue beyond what he already has in terms of the message to the American people," Tillis said. "Now it's about the mechanics."
Lankford said he anticipated something on "paper" from the White House by Tuesday, though lawmakers have been asking for such guidance for weeks.
Democrats, for their part, wave off Republican accusations that they are not being serious on a border security compromise as noise, pressing on in the Durbin-hosted negotiations.
"Anybody who thinks that isn't paying attention or has their own agenda," said a Democratic Senate aide.
Senate Minority Leader Chuck Schumer at a news conference Thursday dodged an opportunity to attack Republicans' demands on "chain migration" and the visa lottery.
"I'm not going to negotiate in front of everyone here," the New York Democrat said. "We've always said we need strong and real border security, not things that sound good but don't do the job. And we need to help the (DACA recipients). That's what we believe, and we will sit down with our Republican colleagues and try to negotiate."
As a January 19 government funding deadline rapidly approaches, Democrats are still insisting a DACA deal must be had but are also continuing to hope negotiations bear fruit, alarming some progressives.
"It's concerning that Schumer and Pelosi are not positioning and framing on this," tweeted
Center for American Progress' Topher Spiro, speaking of House Minority Leader Nancy Pelosi, a California Democrat. "They're not setting themselves up to win public opinion and the blame game."
In December, when Democrats helped Republicans punt the issue to January, a Senate Democratic leadership aide noted that it made no sense to force the issue when negotiations were still productive.
"I can't imagine Sen. Schumer or Ms. Pelosi wanting to shut down the government over this issue when there is a bipartisan commitment to work on it in good faith," Cornyn said Thursday, reiterating that Senate Majority Leader Mitch McConnell had promised Flake he would call a bill for a vote by the end of January if a compromise were reached.
Until then, 60 is the magic number -- the number of votes required in the 51-49-split Senate to advance legislation.
"We got to get to 60, we've got to be reasonable and we've got to get it done," Tillis said Wednesday.