President Donald Trump and congressional leaders, including top Democrats that oppose his proposed southern border wall, will try again Friday to make progress on ending a partial government shutdown. But the odds of a breakthrough appear small.

“Without a wall, you cannot have border security. Without a very strong form of barrier — call it what you will — but without a wall, you cannot have border security,” Trump said Thursday during a surprise visit to the White House briefing room, his first formal appearance there.

“It won’t work,” Trump said flatly, flanked by several border security agents who also spoke in favor of building a border wall. In recent days, the president has not only doubled down on his $5 billion border barrier funding demand to reopen nine Cabinet agencies and other federal offices, but he has increased his desired amount to $5.6 billion.

[Trump congratulates Pelosi in briefing room, stumps for ‘wall’ by any other name]

Hours later, at the Capitol, Speaker Nancy Pelosi was asked if there is any scenario under which she would agree to give the president $1 for his proposed border wall.

Watch: What really happens during a government shutdown, explained

“A dollar? A dollar,” she quipped. “One dollar, yeah, one dollar.”

“The fact is a wall is an immorality. It’s not who we are as a nation. And this is not a wall between Mexico and the United States that the president is creating here. It’s a wall between reality and his constituents and his supporters,” she said hours after reclaiming the gavel.

As the two sides prepared to meet face-to-face for the second time in just three days, the White House was eager to paint Pelosi as beholden to the far-left wing of her party.

“The people that elected Nancy Pelosi didn’t elect her to come up here and do nothing. They didn’t elect her to play political games. They elected her to find real solutions,” White House Press Secretary Sarah Huckabee Sanders told reporters outside the West Wing Friday morning.

“This will be a big test of leadership for Nancy Pelosi. Can she bring the parts of her party together and come up with a solution the president will [support]?” Sanders said. Minutes earlier, during an interview on Fox News, Sanders questioned whether the speaker can “bring the far-left, some of the radical liberals in her party” in line to support a deal with Trump.

For her part, Pelosi on Thursday night made her feelings about the president clear just hours before she will enter the Situation Room for a new round of talks.

“He does not want them to know how he’s hurting them, so he keeps the subject on the wall,” she said of Trump’s conservative core of supporters. “He’s a master of diversion.”

[House Democrats unveil first major legislative package of voting, campaign finance and ethics overhauls]

Former GOP Sen. Rick Santorum on Friday said both sides might have to agree to broaden the deal to include the Deferred Action for Childhood Arrivals, or DACA, program. In short, Trump would get wall funding and Democrats would get at least a partial fix to that program, which they deeply support.

“I think the president feels very little pressure to do anything to keep the government going,” he told CNN. “He feels the imperative is to get the border secure. … The question is whether Democrats can get enough … of their priorities, be it spending or changes in DACA.”

Asked on a chilly White House driveway if Trump might support adding DACA to any coming deal, Sanders did not rule it out. But she said she would let her boss handle the negotiations.

Senate Minority Leader Charles E. Schumer, another participant in the Friday meeting, did not sound particularly optimistic Thursday about progress being made during the resumption of what was a Wednesday Situation Room briefing by Homeland Security officials that failed to produce any progress.

The New York Democrat compared negotiating with Trump to negotiating with “Jell-O.”

Schumer was asked by reporter about whether Majority Leader Mitch McConnell should play a more active role in the negotiating. “He’s the leader of the Senate,” the Senate’s top Democrat said. “When he just tosses the ball over to Trump, he’s somewhat complicit in the shutdown.”

New Senate Majority Whip John Thune on Thursday said he hoped a resolution could be reached soon — but, like every other official involved, admitted he does not yet see a path forward.

“It seems like the sides are kind of far apart, but at some point, you know, the Democrats have got to come to the table and be realistic and reasonable,” the South Dakota Republican said. “I think the president’s flexible, but he’s got to have a dance partner.”

Niels Lesniewski and Lindsey McPherson contributed to this report.