The Colombia international, who joined the Blues for £23.3m from Fiorentina in February, has been linked with a loan move to the Italian champions.
The 27-year-old has made only 15 appearances for the Stamford Bridge side, including four starts.
The Premier League champions signed Spain winger Pedro from Barcelona for £21.1m on Thursday.
On his arrival in Turin, Cuadrado was greeted by waiting fans and the club posted a video saying: “He has arrived! He quickly found an extraordinary welcome. #welcomecuadrado”
Via BBC Sport