His companion, Nancy Beffa, said the cause was complications from Parkinson's disease, the New York Times reported.
Known for his uplifting films on athletes who participated in the Winter and Summer Olympics, Greenspan also made documentaries about baseball and boxing greats.
"I don't do controversy well," he once said.
"I wouldn't know how to handle it.
"I'm thankful I'm not involved in the bureaucratic commercial side of the Olympics."
He also branched out and made documentaries about the baseball player Larry Doby and boxing greats, as well as a made-for-TV movie about the Olympic sprinter Wilma Rudolph.
In his early 20s, Greenspan was the sports director of WMGM Radio in New York and was a part of the Metropolitan Opera chorus.
He eventually shifted his focus to the Olympics, with works like "Jesse Owens Returns to Berlin" and a 22-part series, "The Olympiad," that led to assignments to make the official films of numerous Olympics.
Greenspan received the Olympic Order award in 1985, at which time International Olympic Committee President Juan Antonio Samaranch said: "Mr. Greenspan has been called the foremost producer, writer and director of Olympic films; more than that, he is an everlasting friend of the Olympic family."
In 2004, Greenspan was inducted as a "Special Contributor" into the United States Olympic Hall of Fame.
His visual and musical The Spirit of the Olympics is on display permanently at the Olympic Museum in Lausanne.
Scott Blackmun, the chief executive of the United States Olympic Committee, led the tributes to Greenspan.
"For more than six decades, Bud Greenspan has connected the Olympic Movement to everyday people in ways the founders of the Games couldn't have imagined. Through his films, we came to truly know the Olympic athlete and unique spirit that lives in each competitor.
"On behalf of the US Olympic family, I'd like to express my condolences to Bud's partner, Nancy Beffa, and to the inestimable number of people the world over who called Bud their friend," he said.
"While the entire Olympic Movement mourns the loss of a giant today, his stories will continue to live on, carrying his passion for the Olympic Movement for years and years to come."