“I don’t know how I’m going to do this,” the operator said with a heavy sigh. She suggested she would contact Mr. Trump’s assistant, but then decided to try to transmit the call “through signal,” a reference to the White House Communications Agency, originally known as the White House Signal Corps, which provides emergency mobile communications for the president wherever he goes.

The next audible voice is the signal officer, who can be heard telling Mr. Melendez that the president was onstage at his rally in Fargo, and would have to call back.

When he was a private citizen, Mr. Trump frequently called in to the shock jock Howard Stern’s bawdy radio program — the same one that made Stuttering John, Mr. Stern’s sidekick on the show for more than 15 years, famous — and engaged in salty banter. But Mr. Melendez was plainly shocked at the president’s willingness to take his call in this case.

“I don’t even think they’re going to call me,” Mr. Melendez told his producer at one point, as he as he waited for the call from Mr. Trump.

The White House operator did call Mr. Melendez back at one point to verify that he was who he said he was, asking whether he was calling from a cellphone and why it appeared to be from a California area code. Posing as Mr. Moore, the senator’s aide, Mr. Melendez said he was on vacation.

Later, Mr. Melendez said that he received a call from Jared Kushner, the president’s son-in-law and senior adviser, arranging a callback time for him and the president and asking what topic he would like to raise with Mr. Trump.

When a mobile communications officer did call back and put Mr. Trump on the line, Mr. Melendez played the part of the senator, although his voice sounded nothing like Mr. Menendez’s. But he barreled forward as he engaged Mr. Trump in a discussion about what he should say to his constituents in New Jersey about the Trump administration’s immigration policies that have led to migrant families being separated at the southwestern border.