United said Foulkes passed away in the early hours of Monday morning.
Foulkes made his debut for United in 1952 and went on to play 688 times for the club - a figure only surpassed by Ryan Giggs, Sir Bobby Charlton and Paul Scholes.
Manchester United's executive vice-chairman Ed Woodward paid tribute to Foulkes, saying: "Bill was a giant character in the post-war history of Manchester United.
"He was a very gentle man, who I was privileged to meet on several occasions, including most memorably with his team-mates at the Champions League final in Moscow, 50 years after his heroics in the Munich air crash.
"Bill's contribution over almost 700 games and nearly 20 years will never be forgotten. The thoughts of everyone at the club - directors, players, staff and fans - are with Bill's family."
Foulkes, whose father and grandfather had both played rugby league for his home town St Helens, started work as a coal miner and was still going down the pit at Lea Green Colliery in the mid-50s, by which time he was a regular member of Matt Busby's United first team and had won his only England cap, in 1954.
After the Munich tragedy, he took over the captaincy and became the leader of the 'Busby Babes'.
He played for United for his whole career, the highlight coming near the end of his playing days when he was part of the 1968 European Cup-winning team, aged 36. Foulkes had played a key part in the semi-final too, scoring the winning goal against Real Madrid at the Bernabeu.