In the swirl of money and sun-tanned women that was their Palm Beach-and-Manhattan set, Donald J. Trump and Jeffrey Epstein spent nearly 15 years mingling side-by-side as public friends.

There were lavish dinners with boldface names at Mr. Epstein’s mansion on the Upper East Side and raucous parties with cheerleaders and models at Mr. Trump’s private club and residence at Mar-a-Lago. In between, there were trips back and forth from Florida to New York on one of Mr. Epstein’s private jets.

But behind the tabloid glamour, questions have lingered about what Mr. Trump’s long association with Mr. Epstein says about his judgment and character, especially as his allies have stoked sinister claims about Mr. Epstein’s connections to Democrats. After their relationship ruptured, the disgraced financier ended up behind bars not just once, but two times, after being accused of engaging in sex with teenage girls.

One of the young women who later said Mr. Epstein groomed and abused her was recruited into his world while working as a spa attendant at Mar-a-Lago. Another accuser recalled being eyed by Mr. Trump during a brief encounter in Mr. Epstein’s office, and claimed that Mr. Epstein had told Mr. Trump at the time that “she’s not for you.”

Another woman has said that Mr. Trump groped her when Mr. Epstein brought her to Trump Tower in Manhattan to meet him. This week, The Wall Street Journal reported that Mr. Trump gave Mr. Epstein a note for his 50th birthday in 2003 that included a sketch of a naked woman and a cryptic reference to a “secret” the two men shared. Mr. Trump has denied writing the message and filed a libel lawsuit on Friday challenging the story. The New York Times has not verified the Journal report.

Mr. Trump has never been accused of wrongdoing in connection with the Epstein case, and has said he had “no idea” that Mr. Epstein was abusing young women. In response to a request for comment about the president’s history with Mr. Epstein, Karoline Leavitt, the White House press secretary, said that Mr. Trump had barred Mr. Epstein from his Mar-a-Lago club “for being a creep.”