Greenhoff, who was born in Barnsley, joined United in 1968 and played in the same team as his brother Jimmy, who joined the club in 1976.
The central defender, capped 18 times for his country, scored 17 goals in 271 appearances for United and was part of the victorious FA Cup final team of 1977.
He joined Leeds United in 1979 but left when they were relegated three years later.
After brief spells in South Africa and Finland he returned to play for and coach Rochdale before calling time on his football career in 1983.
Manchester United said on their website:"Everyone at Manchester United is shocked and deeply saddened by the news. Our thoughts and prayers are with Brian's family at this difficult time."
Earlier a statement had been released to the Manchester Evening News by Greenhoff's family that said: "We regret to inform of Brian's passing this morning.
"Brian was a proud and much loved brother, husband of Maureen, father of Paul, Brian and Peter, grandfather to Jack, James and Harry and will be sorely missed by his family, who he made very proud in his distinguished life and career."