"Football loses an extraordinary professional, Torino weeps for a friend. Ciao Gigi, we'll never forget you," Torino said in a statement, lamenting the passing of "a pillar of our history".
In 1976 Radice guided Torino to their seventh Serie A title and first after the Superga disaster in 1949 when a plane carrying the entire Torino team -- the reigning Italian champions -- crashed on the outskirts of the city with no survivors.
Torino had won five of their seven Serie A titles in the 1940s and struggled for decades after the tragedy.
Radice was credited with introducing to Italy the 'total football' concept made famous by the Dutch national football team and Johan Cruyff with Barcelona.
"He was an extraordinary innovator and a maestro, a guide for generations of coaches," Torino said.
As a player, left-back Radice won three Serie A titles with AC Milan and helped them become Italy's first-ever European Cup champions in 1963. He was also capped five times for Italy.
After retiring as a player he turned to coaching, leading Italian teams including AC Milan, AS Roma, Bologna and Fiorentina.
Radice took over Torino in 1975 and won the title the following season. They lost the title by one point in 1977 to city rivals Juventus.
He left the club in 1980 but returned again between 1984 and 1990. Torino finished runners-up in 1985.
Former Torino midfielder Claudio Sala, 71, who captained the side to the 1976 trophy, praised "a great coach who changed Italian football".