Robson had been fighting cancer for a number of years.
He played for Fulham and West Brom during the 1950s and 1960s, and later went on to manage Fulham, Ipswich, PSV, Sporting Lisbon, Porto, Barcelona and the club he supported as a boy, Newcastle.
A statement issued on behalf of his family said: "It is with great sadness that it has been announced today that Sir Bobby Robson has lost his long and courageous battle with cancer.
"He died very peacefully this morning at his home in County Durham with his wife and family beside him.
"Sir Bobby's funeral will be private and for family members only.
"A thanksgiving service in celebration of Sir Bobby's life will be held at a later date for his many friends and colleagues.
"Lady Robson and the family would very much appreciate it if their privacy could be respected at this difficult time.''
Last year he launched the Sir Bobby Robson Foundation to raise money for cancer research.