WASHINGTON — President Trump on Tuesday transformed what was to be a private negotiating session with Democratic congressional leaders into a bitter televised altercation over his long-promised border wall, vowing to force a year-end government shutdown if they refused to fund his signature campaign promise.

During an extraordinary public airing of hostilities that underscored a new, more confrontational dynamic, Trump vowed to block full funding for the government if Democrats refuse to allocate money for the wall on the Southwestern border, saying he was “proud to shut down the government for border security.’’

Senator Chuck Schumer of New York and Representative Nancy Pelosi of California, the Democratic leaders, took issue with the president’s position and his false assertions about the wall in front of news cameras, imploring him repeatedly to continue the tense conversation without reporters present. But Trump insisted on a conspicuous clash that undercut Republican congressional leaders and his own staff working to avoid a shutdown at all costs, or at least to ensure that Democrats would shoulder the blame for such a result.

“If we don’t have border security, we’ll shut down the government — this country needs border security,’’ Trump declared as the testy back and forth unfolded.

Schumer reminded the president repeatedly that he had called several times for a shutdown, appearing to goad him into taking responsibility.

“You want to know something?’’ Trump finally said, exasperated. “I’ll tell you what: I am proud to shut down the government for border security, Chuck. I will take the mantle. I will be the one to shut it down — I’m not going to blame you for it.’’

The scene raised fresh questions about whether Trump and lawmakers could reach an agreement by a Dec. 21 deadline to keep much of the government open.

Later at the Capitol, Pelosi spared little venom in privately describing the president’s demeanor to colleagues. “This wall thing’’ of Trump’s is “like a manhood thing for him — as if manhood could ever be associated with him,’’ Pelosi told fellow Democrats, according to an aide present for the remarks who insisted on anonymity.

Pelosi and Schumer also sparred with Trump about the financing of the wall, after the president insisted Mexico would pay for it through money that would flow into the United States as part of a new North American trade agreement. The Democrats told Trump that would never happen. Pelosi said she had told him that any economic gains from the pact should go to US workers and businesses, not a border wall.

“Well, they’re going to pay for it one way or the other,’’ Trump insisted, said Pelosi.

After the meeting, Schumer said Trump had thrown a “temper tantrum,’’ later telling reporters, “You heard the president: He wants a shutdown.’’

Yet Trump appeared to be open to a way out of the impasse. He called Pelosi hours after the meeting, she said, and was reviewing options Democrats had presented. Trump even went into a speech during the Oval Office session, once reporters had departed, in which he mused about the potential for him to work alongside Democratic leaders in the new Congress to strike great deals.

Trump told Schumer and Pelosi the new Congress could be the “greatest Congress in the history of Congress,’’ said a person familiar with the discussion who described it on the condition of anonymity because it was private. And the president assured them that if they could strike deals that could get through the House, he would compel Republican senators to back them. On Tuesday, though, such compromises were not evident. Trump’s outburst left Republican congressional leaders in a tricky spot, rejecting a shutdown that their president had promised.

“I hope that’s not where we end up,’’ Senator Mitch McConnell, Republican of Kentucky, the Senate majority leader, told reporters. “It was a rather spirited meeting we all watched, but I’d still like to see a smooth ending here.’’

Veteran legislators expressed outrage.

“This is the only time I’ve seen anything as irresponsible as a president who says he wants a government shutdown,’’ said Senator Patrick J. Leahy, Democrat of Vermont.

In morning tweets, Trump falsely stated that substantial sections of the “Great Wall’’ had been completed, and suggested he could continue construction whether Democrats fund it or not. That would be illegal, but it suggested he was looking for a way to keep the government funded past Dec. 21.