John Melendez, who is better known by his longtime moniker on "the Howard Stern Show," Stuttering John, released audio of the purported conversation on his latest podcast Thursday.
In the call, Melendez poses as Sen. Robert Menendez, D-N.J., and appears to be patched through to Trump while the president is aboard Air Force One on his way back to Washington after his Wednesday night rally in Fargo, North Dakota.
The White House and Menendez's office did not immediately respond to requests for comment.
"Hi, Bob," Trump says on the call, according to the audio aired on "the Stuttering John Podcast," before going on to offer his congratulations to "Menendez." The New Jersey Democrat fended off a challenger this month in a closer-than-expected Senate primary after avoiding conviction in a federal corruption trial last year.
"You went through a tough, tough situation - and I don't think a very fair situation - but congratulations," Trump says on the call.
Melendez and Trump then have a chat about immigration policy and the process of choosing a Supreme Court nominee before wrapping up their conversation a few minutes later.
"You take care," Trump says at the end of the call. "I will speak to you soon, Bob. Take care of yourself."
The news comes one month after reports that Trump's use of government-issued smartphones may open him up to security risks. The president also faced criticism last year for discussing the sensitive issue of how the U.S. should respond to a North Korean ballistic missile test while on the terrace of his Mar-a-Lago Club in Palm Beach, as waiters and curious patrons watched and snapped photos.
Melendez, who left Stern's show in 2004 and went on to work as an announcer for Jay Leno's "Tonight Show," has long been known for ambushing celebrities with embarrassing questions. In one memorable appearance, he asked Gennifer Flowers at her 1992 news conference about her affair with Bill Clinton, "Will you be sleeping with any other presidential candidates?"
"A lot of these celebrities don't get it - they don't want to ever be able to laugh at themselves," Melendez told The Washington Post in a 1994 interview. "They are so used to being treated as the elite, they put themselves on such a pedestal. They forget they are just human beings."
In a tweet Thursday night, Melendez said that he "totally duped the President" and had gotten in touch with him within "less than 2 hours while he was on Air Force One." He also voiced surprise that news outlets had not picked up the story.
On the podcast, Melendez first calls the White House switchboard and identifies himself as "Stuttering John Melendez," but is told that Trump is unavailable. He then calls back and identifies himself as Menendez's assistant and says that the senator wishes to speak to Trump about "a certain bill that's pressing right now."
A White House operator eventually calls back and connects him with Trump aboard Air Force One.
In the conversation, Melendez asks Trump how he plans to proceed on the hot-button topic of family separations at the U.S. southern border, telling the president, "My constituents are giving me a lot of beef about this immigration thing."
Trump replies by voicing optimism that Congress can "do a real immigration bill." Lawmakers left town this week after failing to pass two separate immigration bills, and they remain divided on how to proceed on a narrower solution to the family-separation issue.
"I'd like to do the larger solution rather than the smaller solution," Trump says on the call, adding: "I think we can do the whole thing. I have a good relationship with the party; you have a good relationship with the party."
He goes on to advocate for stronger border security, telling Melendez that it's "good for both of us."
"No, I understand that, but I am Hispanic," Melendez replies. "So, you know, I have to look after my people as well. You understand."
"I agree. I agree," Trump says.
Melendez also tells Trump on the call not to "go too conservative" with his choice for a Supreme Court nominee to replace Justice Anthony M. Kennedy, who announced his retirement Wednesday.
Trump responds that he expects to make a decision on a nominee "over the next 12 to 14 days."
"I have a list of people. I have a big list of people, Bob, and we'll take a look and it and we're going to make a decision," Trump says.