The 23-year-old Italy international won three consecutive Serie A titles with the Bianconeri in his five years in Turin.
He began his career with a two-year loan spell at Empoli, culminating in being named 2014 Serie B Player of the Year, followed a year later by inclusion in the Serie A Team of the Year.
His arrival exacerbates Chelsea’s defensive problems, not least of which is the future of England international Gary Cahill, 32.
The centre-back grew frustrated last season when Antonio Conte preferred Antonio Rudiger on the left side of the back three. His situation is unlikely to improve with new head coach Maurizio Sarri preferring a back four; Chelsea are overloaded with defenders who can play centrally.
Cahill is yet to return to pre-season training following his involvement in the World Cup finals with England. He is expected to meet with Sarri to discuss his future after a six-year spell at Stamford Bridge.
The situation is clouded by the continued presence in the squad of David Luiz and Andreas Christensen in the squad. Rugani’s arrival will increase the number of specialist centre-backs to five with Cahill’s desire to make more than the 24 Premier League starts of last season looking increasingly unlikely.