House Speaker Nancy Pelosi, D-Calif., sidestepped a question about the issue Thursday, telling reporters, "If and when the president does that, you'll find out how we will react."
She predicted that if the president did take that step, he would have to answer to his own party for "usurping" power.
Trump on Thursday reiterated that Republicans are unified in support of him on the wall. He hasn't "ever seen unity like this in the Republican Party," he said.
However, several Republican senators have expressed concern over the effects of the continued partial government shutdown, now in its 20th day. A handful of House Republicans have also voted with Democrats in efforts to reopen the government.
Trump also disputed characterizations of Wednesday's meeting with Democratic leaders, in which Trump abruptly walked out. "I didn't pound the table," he said. "I didn't raise my voice. That is a lie."
He added, "I don't have temper tantrums. I really don't."
Trump also said that despite his repeated campaign promises to build a wall and have Mexico pay for it, he didn't mean that literally.
"When I said Mexico would pay for a wall in front of thousands and thousands of people," Trump said, "obviously I never meant Mexico would write a check."
But a March 2016 memo from the Trump campaign to The Washington Post said, "It's an easy decision for Mexico. Make a one-time payment of $5-10 billion to ensure that $24 billion continues to flow into their country year after year."
In Pelosi's news conference, she suggested that Wednesday's White House meeting with Hill leaders was never about a serious negotiation. "I think the meeting was a setup so he could walk out," she said.
She said the president keeps changing his demands: "I don't even know if the president wants the wall," Pelosi said. "I said to him 'You are moving the goal posts so many times they are out of the stadium.' "
Democratic congressional leaders have refused to go along with Trump's demand for border wall funding, and Senate Majority Leader Mitch McConnell has refused to take up Democratic measures to fund government agencies that have run out of appropriations.
Some 800,000 federal workers have been furloughed or are working without paychecks because of the standoff. Many will be missing their first paychecks on Friday.