Newsmax, another media outlet favored by the president, reported that Mr. Hannity explicitly said Mr. Trump was now willing to negotiate the fate of the Dreamers to secure his wall.
But Mr. Trump is also facing pressure from his base to deliver on the wall, a central promise of his presidential campaign. Mr. Graham said Wednesday in an interview on Fox News that Mr. Trump’s presidency could effectively be ended if he abandoned the wall pledge. And the president this week rejected the “Dreamers for wall” idea.
On Thursday, Mr. Trump made a dramatic appearance in the White House Briefing Room, his first as president, trying to press his case for the wall by asking border agents to outline the threats posed by illegal immigration. But he simply reiterated his position and took no questions.
Senator Mitch McConnell of Kentucky, the majority leader, would typically be a central player in the negotiations. So far, though, he has held back, contending that it is up to Democrats to fashion a solution that the president could accept.
The House has already passed a two-bill package to reopen the government. The first measure combines six separate bills that have already garnered bipartisan support in the Republican-led Senate; they would reopen nearly all of the shuttered agencies and fund them through Sept. 30, the end of the fiscal year.
The second is a stopgap spending measure to fund the Department of Homeland Security through Feb. 8 — a date that Mr. McConnell proposed toward the end of last year in a measure that passed the Senate by voice vote, but which the president rejected, triggering the partial shutdown.
Mr. McConnell has refused to take up the House package, insisting that he will not bring anything to the floor that Mr. Trump will not sign. Democrats intend to argue in the meeting that Mr. McConnell should at the least pass the cluster of appropriations bills, while continuing to negotiate over border security. In their last meeting, Mr. Trump rejected that idea, telling the group, “I would look foolish if I did that.”
Mr. McConnell, meanwhile, is waiting for guidance from Mr. Trump. The Senate Republican leader has absented himself from the talks, insisting that it is up to Democrats to resolve the impasse. But he is beginning to face pressure from vulnerable Republicans who are worried about their re-election prospects in 2020.