In her opening comments in the Oval Office, Pelosi said failed budget talks would result in a "Trump shutdown" of government.
"If the Democrats do not give us the votes to secure our Country, the Military will build the remaining sections of the Wall", Trump said on Twitter ahead of the 11:30 a.m. EST (1630 GMT) meeting. He said he would shut down the government if "we don't get what we want".
The California Democrat said in a statement, "Over the summer, I made it clear that I see myself as a bridge to the next generation of leaders, a recognition of my continuing responsibility to mentor and advance new Members into positions of power and responsibility in the House Democratic Caucus".
"I will take the mantle, I will shut it down", Mr Trump said.
"This temper tantrum that he seems to throw will not get him his wall and it will hurt a lot of people", he said. "I can't explain it to you".
Under the deal, House Democrats will vote by February 15 to change party rules to limit their top three leaders to no more than four two-year terms, including time they've already spent in those jobs.
After January 2021, they'd need the two-thirds caucus vote to stay in place, Politico said.
Democrats widely agreed that the pledge meant Pelosi had clinched a comeback to the post she held from 2007 until January 2011, the last time her party ran the House and the first time the speaker was a woman.
Winter storm slams large part of U.S.
Major snow and ice will cause travel impacts across a widespread area.Ice accumulations could cause power outages and tree damage. A weather system the National Weather Service is calling "complicated" will arrive in Arkansas beginning Friday.
Trump earlier threatened a partial government shutdown in case a deal isn't reached by December 21, but Democrats have passed a two-week stopgap spending bill in order to avert such a scenario.
"It's like a manhood thing for him", Pelosi told Democratic members at a closed meeting of the Steering and Policy Committee on Capitol Hill, according to an aide in the room.
Trump's lawyer and former New York City mayor Rudy Guiliani told the Times that he would tell the president to fight any request for his tax returns.
MSNBC's Ali Velshi, a daytime host who occasionally fills in for the network's primetime pundits, said on "All In" that Pelosi showed House Democrats preparing to elect their next speaker of the House that she was the right pick.
When Pelosi brought up Republican election losses in the House, Trump quickly retorted that his party won the Senate.
CNN anchors Chris Cuomo and Don Lemon chuckled together about "NanChuck" taking on Trump over the wall.
"It's not bad, Nancy - it's called transparency", Trump said.