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Don, the pudgy old hedge fund manager whose fuzzy bearlike face has been a fixture of Trump Tower’s elevators since the 1980s, may be out of a job.

A source confirmed to The New York Times that the president-elect will meet with brokerage-house owner Herbert Allen Jr. on Friday morning, and let him pick his replacement on the 65th floor.

Two advisers to Mr. Trump said it was a possibility that Mr. Trump, who spent much of his campaign denouncing Wall Street executives and bankers as the “global elite,” could look elsewhere from here on out.

“The president-elect knows he’s going to pick someone from a different background,” said John Catsimatidis, a supermarket magnate and frequent political adviser to Mr. Trump, who declined to name the candidate. “Because Don is not there any more.”

Mr. Trump owns the two adjacent towers on the 16th and 17th floors of Trump Tower, which occupies the entire block between Fifth and Sixth avenues, all but an elevated portion of which is currently under construction. Trump broke ground in 2003 on a 1,300-foot tower across the street that will offer only 360-degree views.

Richard A. LeFrak, a principal in the LeFrak Organization, was in town Friday from Las Vegas to discuss the project. Mr. LeFrak, the son of the real estate magnate, told The New York Times in November that the skyline-shattering building could be ready in April for its first guests.

“So what we’re going to do is talk about our plans and what we’re doing,” he said. “I think we’ll talk about the [Trump] towers.”

The president-elect is planning to announce his nomination of Mr. Trump’s longtime friend and New York lawyer Rudolph W. Giuliani as secretary of state shortly after Labor Day. The Times reported earlier this week that they are unlikely to pick a deputy secretary of state before Mr. Giuliani’s nomination is unveiled.

The president-elect will also meet in two weeks with an acquaintance of his from his days as a New York City business mogul, Andy Puzder, to discuss the labor secretary post. Mr. Puzder, the CEO of CKE Restaurants, has voiced support for replacing most jobs with robots.

“He’s not a political person, and he’s in that same boat as most New Yorkers — no political friends in Washington,” said Stephen Steiner, a prominent businessman who is close to the Trump family. “He’s not going to spend a lot of time in D.C.”