A row with Democrats over funding the wall has left the government partially shut down for 20 days, leaving some 800,000 federal employees unpaid.
On Saturday, the shutdown becomes the longest in US history.
Mr Trump says the wall, a key campaign pledge, is needed to tackle a security crisis of illegal immigration.
The Democrats say the wall is an "immorality" and a waste of taxpayers' money.
President Trump has refused to sign legislation to fund and reopen the government if it does not include $5.7bn (£4.5bn) for the wall.
What's behind Mr Trump's threat and what could happen?
He issued the threat on a visit on Thursday to a border patrol station in McAllen, in the Rio Grande Valley of Texas.
He said that if Congress did not approve funding for the wall, he would "probably... I would almost say definitely" declare a national emergency to bypass lawmakers.
"I have the absolute right to declare a national emergency," he said.
Mr Trump would have the right to undertake such a construction project in times of war and national emergency, usually allocating funds from the department of defence.
But bypassing Congress would be hugely controversial, sparking allegations of the overuse of executive powers and it would almost certainly face huge legal challenges.
Some US media reports suggest the White House is considering diverting some of the $13.9bn (£10.9bn) allocated last year by Congress for disaster relief in such areas as Puerto Rico, Texas and California to pay for the wall.
NBC News reported on Thursday that Mr Trump had been briefed on such a planbut the White House denied it.
Analysts say the national emergency move would provide political cover to reopen government while allowing Mr Trump to argue he has done all he can to fulfil his campaign promise.
The BBC's David Willis in Washington says that, with no further talks with the Democrats planned, this now seems the most likely option for the president.
Republican Senator Lindsey Graham, a key Trump ally, said there was no other pathway forward.
What did Mr. Trump say at the border?
He was speaking at McAllen station, behind a display of weapons and cash said to have been seized by the border patrol.
He was joined by border patrol agents and relatives of people killed by illegal immigrants.
"If we don't have a barrier... you're not going to be able to solve this problem," he said, adding that people faced "hard work", "gruelling problems" and "a lot of death" without it.
Mr Trump added: "They say a wall is medieval... There are some things that work."
The president said in a national address on Tuesday the wall was needed to stem a "growing humanitarian and security crisis at our southern border", involving "thousands of illegal immigrants"
He pointed to illegal drug supplies, people trafficking and criminal acts by illegal immigrants in the US.
The Democrats say the wall is "ineffective" and "unnecessary" and an expensive bill to taxpayers that the president had said Mexico would foot.