Scotland's joint-leading scorer had been diagnosed with dementia in 2021.
"Everyone at Manchester United is mourning the loss of Denis Law, the King of the Stretford End," United said in a statement on Friday.
The club added he will "always be celebrated as one of the club's greatest and most beloved players.
Man United's Denis Law (r) and Fulham's Bobby Robson give chase during their clash in 1966. (AP PHOTO)
"The ultimate goal-scorer, his flair, spirit and love for the game made him the hero of a generation."
Law scored 237 goals in 404 appearances for United and was part of their acclaimed "Holy Trinity" alongside Bobby Charlton and George Best in the all-conquering team of the 1960s.
He was the only Scottish player to win the Ballon d'Or.
Law also played for United's fierce local rival Manchester City, including his last season in 1974.
Manchester City said the "whole of Manchester" is in mourning.
In a reply to United's X post breaking the news, they said: "The whole of Manchester, including everyone at City, is mourning with you.
"Rest in peace, Denis. Our thoughts are with Denis' family and friends at this difficult time."
Merseyside arch-rivals Liverpool posted on X: "Denis was a rival player but so highly thought of. As (former manager) Bill Shankly once said, 'Denis Law could dance on eggshells."
Manchester United great Bryan Robson, who worked with Law in an ambassadorial role for the club, called the Scot a "fantastic man".
"We have lost one of football's giants both as a player and a gentleman. It's incredibly sad," he said in a column in the Daily Mail.
"Denis was more than just a fantastic footballer, he was a fantastic man. So generous with his time and everything delivered with that great sense of humour of his.
"He would always be in my greatest ever Manchester United XI. He was a player so many of his peers idolised and with good reason, that iconic image of him with his sleeves pulled down and the one arm salute after scoring."
Law's club career began at Huddersfield and the West Yorkshire club called him an "immortal of the sport".
They said in a statement posted on X: "All at Huddersfield Town are profoundly saddened to learn of the passing of Denis Law.
"A legend of not only our great club, but an immortal of the sport as a whole, he will be deeply missed and his memory cherished by us all."
Scotland's men's national team X account added: "A true great. We will not see his likes again."
Former United captain Gary Neville added on Instagram: "Rest in Peace Denis. A great footballer and a great man. It's a privilege and an honour to have spent time in your company. The King of the Stretford End."
Manchester United's record goalscorer Rooney added on X: "Legend. Thoughts with all Denis's family and friends."
Former Scotland international Joe Jordan said on BBC Radio 5 Live: "A proper icon for his ability on the field to score goals but he had a special edge to him, not just as a player but as a person."
"On behalf of European football," UEFA posted on X, "we are deeply saddened to learn of the passing of Denis Law. One of Scottish football's true greats, he was a Ballon d'Or winner in 1964 before lifting the European Cup with Manchester United in 1968. Rest in peace, Denis."
with PA
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