“What he got was access to Fred Trump,” Elysa Braunstein said. “If there was anything wrong in the building, my dad would call and Trump would take care of it immediately. That was the small favor that he got.”

No paper evidence has been found to help corroborate the version of events described by the Braunstein family, who also suggested there was some involvement by a second podiatrist, Dr. Manny Weinstein. Dr. Weinstein, who died in 1995, lived in two apartments in Brooklyn owned by Fred Trump; city directories show he moved into the first during the year Donald Trump received his exemption.

Dr. Braunstein’s daughters said their father left no medical records with the family, and a doctor who purchased his practice said he was unaware of any documents related to Mr. Trump. Most detailed government medical records related to the draft no longer exist, according to the National Archives.

In an interview with The Times in 2016, Mr. Trump said that a doctor provided “a very strong letter” about the bone spurs in his heels, which he then presented to draft officials. He said he could not remember the doctor’s name. “You are talking a lot of years,” Mr. Trump said.

Image The yearbook photo of Dr. Manny Weinstein, who may have also played a role in the medical exemption, according to Dr. Braunstein’s daughters.

But he suggested he still had some paperwork related to the exemption, which he did not provide.

Mr. Trump did not mention in that interview any connection between his father and the doctor. The White House did not make Mr. Trump available for a follow-up interview and did not respond to written questions about his service record.

An investigation by The Times in October showed the extent to which Fred Trump had assisted his son over the years, despite Donald Trump’s insistence to the contrary. The investigation revealed that Mr. Trump received the equivalent today of at least $413 million from his father’s real estate empire, including the equivalent of $200,000 a year by age 3.