Sandy Busby, who was in his late 70s, reportedly died on Monday evening at around 7.15pm.
United's executive vice chairman Ed Woodward told the club's official website: "I am shocked and saddened by news of Sandy's death.
"He was such a genuine person who had a deep and rich love for the club that his father did so much to establish in the hearts of millions around the world.
"Sandy understood better than almost anyone what the Manchester United way means and his enthusiasm, his wit and his affection for the club will be sorely missed. Our thoughts go out to Irene and his family at such a sad time."
Sandy's father managed United between 1945 and 1969, and a second spell for one season in 1970.