Mr Trump is trying to convince the Democrats to approve around $5bn (£3.95bn) to build the wall, which he says is vital to stop illegal immigrants and criminals entering the US.
But Democrats do not appear moved by the president's message.
"Look, if Harry Truman couldn't nationalize the steel industry during wartime, this President doesn't have the power to declare an emergency and build a multibillion dollar wall on the border".
Last week, House Democrats proposed a bill to fund the Department of Homeland Security, which oversees construction of the wall, at current levels through February 8, with $1.3 billion for border security-a figure far less than Trump requested.
Mr Trump also visited the southern border last March.
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Trump further said Sunday he was considering whether to bypass Congress and build the wall by declaring a national emergency ― which would constitute an unprecedented use of executive powers.
The president's visit will come on what will likely be the 20th day of the partial government shutdown.
Democrats returned to Washington in the new year, taking control of the US House of Representatives, and passed legislation to reopen all closed government agencies but did not include wall funding.
For instance, during a news conference on Friday, Trump claimed that predecessors in the White House had told him that they would have supported a border wall, but that was contradicted by spokespersons for the living ex-commanders in chief. "So, that's a nonstarter", said Rep. Adam Schiff (D-California), one of Trump's most vocal critics.
She called the debate over using steel versus concrete "bizarre". "He's not building a wall anymore, ' that should help us move in the right direction". Later on CNN, however, Smith said that declaring an emergency would be "horrible policy" and a "terrible idea", and that the president would be challenged in court if he sought to build the wall without congressional authorization.
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At the White House, on Monday, spokeswoman Mercedes Schlapp complained that Democratic leaders have yet to define what they mean when they say they are for enhancing border security. "The president has asked every single Cabinet secretary and the Office of Management and Budget to go out and find money that can be used legally to guard the southern border, which is exactly what we're going to do".
President Trump is considering the possibility of declaring a national emergency, according to Pence, who said they are also considering what Congress can do. "They don't like concrete, so we'll give them steel".
Trump said Sunday a national emergency declaration would depend on how negotiations to break the impasse with House and Senate Democrats over border wall funding play out in the next few days.
New House speaker Nancy Pelosi will this week start passing bills to reopen government agencies, starting with the treasury in an attempt to ensure people receive tax refunds threatened by the government closure. The bills are almost identical to legislation designed by Republicans in the Senate in 2018. But the Republican-controlled Senate seems unlikely to take that step.
Trump's recent remarks about a possible emergency declaration appear to reiterate remarks he made days ago at a January 4 open press conference in the Rose Garden.
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"Walls work", the president wrote in a letter to lawmakers last week.
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