Trump's comments came as he opened a contentious meeting with Democratic congressional leaders at the White House in an effort to avert a partial government shutdown on December 21, when funding for some agencies is scheduled to expire. "We always knew a vote for any Democrat was a vote for Nancy Pelosi's San Francisco agenda, and today's move by Tim Ryan proved just that".
Prior to a meeting with Democratic congressional leaders, Trump told reporters that he would be "proud" to shut down the government if he does not get funding for his proposed border wall. "I will take the mantle. I will be the one to shut it down", he declared.
It was the first time Mr Trump met with Ms Pelosi and Mr Schumer since the Democrats won control of the House of Representatives in the 6 November congressional elections - a rocky start to the relationship the White House will have with the opposition party, with which it needs to deal to advance any priorities.
The president, however, was less enthused by this designation (#TrumpShutdown is now trending on Twitter), retorting, "A what?"
"When you have walls, Chuck, it's effective, when you don't it's not", Mr Trump retorted.
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.
While Pence was also present at the meeting, his noticeable silence has provided for internet fodder.
Schumer said Democrats want to work with Trump to avert a shutdown, but said money for border security should not include the concrete wall Trump has envisioned.
But the harsh words continued during a closed meeting of House Democrats.
At the Pentagon, Lieutenant Colonel Jamie Davis said there were no plans now for the USA military to build sections of the border wall, though the Defense Department could under current law pay for some border barrier projects "such as in support of counter-drug operations or national emergencies".More news: We Could Blow You Away, Liverpool Legend Warns Manchester United
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Hours before the meeting, Mr Trump tweeted that "the Military will build the remaining sections of the Wall".
Pelosi and Schumer have urged Trump to support a measure that includes a half-dozen government funding bills largely agreed upon by lawmakers, along with a separate measure that funds the Department of Homeland Security at current levels through September 30.
"Elections have consequences, Mr. President", said Schumer. "It's his choice to accept one of those options or shut the government down".
The proposal to implement a new rule that would impose term limits was tentatively agreed upon Wednesday by seven members of the Democratic caucus who had said they would oppose her, but lawmakers were waiting on Pelosi's office to officially announce her support for it.
The pact followed weeks of harsh public battles among Democrats and an all-out lobbying campaign by Pelosi, D-Calif., and her allies.
Later in private, she told the Democrats: "I was trying to be the mom".
Pelosi rebutted Trump's claim and said, "There are no votes in the House, the majority votes, for a wall". The Kentucky Republican said "magic" sometimes happens in Congress ahead of Christmas, when lawmakers are eager to leave Washington.
Party rebels have said it's time for Pelosi and her two top lieutenants, No. 2 leader Steny Hoyer of Maryland and No. 3 leader Jim Clyburn of SC, to step aside and make room for a new generation.
Senate Majority Leader Mitch McConnell on Tuesday urged Democrats to accommodate Trump's wall demands, warning lawmakers either needed to cooperate or "prepare for a very, very long month".
Taking out two memo cards, he read off figures that he said showed a near end to illegal immigration at portions of the border already featuring high fences.