Ken Kurson is a confidant of Rudolph W. Giuliani. He is a onetime speechwriter for Donald J. Trump. And he is a close friend of Jared Kushner, Mr. Trump’s son-in-law, who appointed Mr. Kurson to run his weekly newspaper, The New York Observer.

This spring, those relationships appeared to yield a prestigious offer from the Trump White House: a seat on the board of the National Endowment for the Humanities, a federal body that doles out millions of dollars a year in grants to cultural institutions.

First, Mr. Kurson had to undergo a government background check. As part of that process, the Federal Bureau of Investigation learned about allegations that he had harassed a New York doctor in 2015, according to Mount Sinai Hospital, where the doctor worked.

In late spring, the F.B.I. interviewed Mount Sinai doctors and others about the alleged harassment, according to several people familiar with the matter, who requested anonymity to discuss a delicate situation. The female doctor had become so worried that Mount Sinai arranged for someone to accompany her to and from the hospital for at least a few days, two of the people said.