In an exclusive interview aired during Fox News' "All-American New Year" special Monday night, President Trump suggested that only U.S. Sen. Elizabeth Warren's "psychiatrist" knows whether she thinks she can win the White House in 2020.

Warren, a Massachusetts Democrat and progressive firebrand, announced Monday she is filing paperwork to launch an exploratory committee for president, becoming the first candidate to take the major step toward a 2020 run for the presidency.

Fox News' Pete Hegseth asked Trump whether Warren really thinks she could make him a one-term president.

"Well, that I don't know," Trump responded. "You’d have to ask her psychiatrist."

CLICK HERE TO GET THE FOX NEWS APP

Warren, who rose to prominence during the 2008 financial crisis, angered many top Democrats and Native American groups in October by releasing inconclusive DNA test results in response to Trump's claims that she repeatedly lied about her heritage to obtain affirmative-action benefits in the course of her academic career.

The Cherokee Nation responded to the results at the time by asserting that “a DNA test is useless to determine tribal citizenship.” And Kim TallBear, an associate professor at the University of Alberta in Canada, remarked that Warren's "very desire to locate a claim to Native American identity in a DNA marker inherited from a long-ago ancestor is a settler-colonial racial understanding of what it is to be Native American."

"Elizabeth Warren will be the first," Trump told Hegseth in the phone interview. "She did very badly in proving that she was of Indian heritage. That didn't work out too well."

According to Warren's DNA analysis, "the vast majority” of Warren’s family tree is European and there is “strong evidence” she has Native-American ancestry “in the range of 6-10 generations ago.” As reported by the Boston Globe, this means she could be between 1/64 and 1/1,024 Native American.

"I think you have more than she does, and maybe I do too, and I have nothing," Trump said, referring to tribal heritage. "So, we’ll see how she does. I wish her well, I hope she does well, I’d love to run against her."

Trump repeatedly has derided Warren for claiming she has Native American ancestry. At a rally in July, he joked that he would pull out a heritage kit during a hypothetical presidential debate with Warren and slowly toss it at her, "hoping it doesn't hit her and injure her arm, even though it only weighs probably two ounces."

"If you go just based on the record, I don't see how anybody [else] wins" in 2020, Trump said. "It doesn't just seem based on the record that somebody's going to do really well."

The president said the big winners of 2018 are "the American people," owing to progress on the economy and the military. As for the big losers, Trump suggested some of the Democratic candidates lining up against him might soon qualify.

Trump's New Year's resolution: "Success, prosperity, and health for our country."

Separately, Trump again invited top Democrats to join him in Washington to resolve the ongoing partial federal government shutdown -- but he signaled that a border wall is an essential element of any deal.

One bipartisan proposal to end the shutdown that has been floated among key senators is to provide $5.7 billion in funding for the border wall, as well as a congressional reauthorization of the Obama-era Deferred Action for Childhood Arrivals (DACA) program for those brought to the U.S. illegally as children, along with some other immigration provisions. There also has been talk about a special allowance for some classes of Central American refugees to be granted more robust asylum statuses.

"You’d have to ask her psychiatrist." — President Trump on Elizabeth Warren's chances of winning in 2020

"I’m in Washington, I'm ready, willing and able. I'm in the White House, I'm ready to go," Trump said. He added that Democratic House Minority Leader Nancy Pelosi and Senate Minority Leader Chuck Schumer "can come over right now, they could've come over anytime."

The president emphasized that he canceled his plans to spend Christmas and New Year's Day at his Mar-a-Lago retreat in Florida because of the partial shutdown, and signaled that he remains concerned about the approximately 800,000 federal workers who are affected by furloughs and understaffing.

"The wall is not old-fashioned. The wall is one-hundred percent foolproof. ... I spent Christmas in the White House, I spent New Year’s Eve now in the White House," Trump said. "And you know, I'm here, I’m ready to go. It's very important. A lot of people are looking to get their paycheck, so I'm ready to go whenever they want."

He added: "No, we are not giving up. We have to have border security and the wall is a big part of border security. The biggest part."

Later in the interview, Trump reaffirmed his decision to order all U.S. troops out of Syria, but walked back his previous assertion that troops would be completely gone within 30 days.

Following a weekend meeting at the White House with U.S. Sen. Lindsey Graham, R-S.C., who has fervently criticized Trump's pullout, the president made clear to Hegseth that the withdrawal would not be "rushed."

"We're fighting these endless wars. I campaigned on getting out of the endless wars," Trump said. "And frankly I've done more than I've said -- not only have I gotten out, but we've won. ... We have to bring our troops back home. It's time."

Fox News' Chad Pergram and Alex Pappas contributed to this report.