The President is demanding billions of dollars to build a wall along the US border with Mexico, which the Democrats have so far refused.
The disagreement has led to a partial shutdown of the federal government which started on 22 December and has so far affected 800,000 federal workers.
Mr Trump has withheld his support for any new funding package until he secures $5 billion to start building the wall along the US-Mexico border that he promised during his campaign.
Such a wall, he has argued, is needed to stem the flow of illegal immigrants and drugs over the southwestern border.
When he ran for president in 2016, he vowed Mexico would pay for the wall, which it has refused to do.
House Speaker Nancy Pelosi said Mr Trump and Senate Republicans should "take yes for an answer" and approve the border bill to completely end the shutdown.
When asked if she would give Mr Trump one dollar for a wall to reopen the government, Ms Pelosi said: "One dollar? Yes, one dollar. The fact is a wall is an immorality. It's not who we are as a nation."
Ms Pelosi, who was elected speaker earlier on 3 January, also took a shot at Mr Trump, calling his proposal "a wall between reality and his constituents".
The largely party-line votes came after Mr Trump made a surprise appearance at the White House briefing room pledging to keep up the fight for his signature campaign promise.
Reporters were told that White House press secretary Sarah Huckabee Sanders would be holding a late afternoon briefing on 3 January.
Instead, out walked Mr Trump, flanked by members of the unions that represent border patrol and immigration enforcement agents.
It was his first time delivering remarks at the briefing room podium.
"You can call it a barrier, you can call it whatever you want," Mr Trump said.
"But essentially we need protection in our country. We're going to make it good. The people of our country want it."
Mr Trump said his meeting with the union officials had long been planned and just happened to come at "a very opportune time".
He also claimed his refusal to budge was winning praise, telling reporters, "I have never had so much support as I have in the last week over my stance for border security."
He then left without taking questions from reporters.
The Democratic legislation to re-open the government without funding the wall is going nowhere in the Senate, where Republicans want Mr Trump's endorsement before voting on a funding package.
Congressional leaders from both parties met with Mr Trump at the White House on 2 January, but failed to make progress during their first sit-down in weeks.
The White House has invited the leaders back on Friday for another round of talks that officials have suggested might be more successful now that Ms Pelosi has been sworn in.
Polls show a majority of Americans oppose the border wall, although Republicans strongly support it.
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