The Colombian, who moved to Stamford Bridge from Fiorentina for £23.3m in February 2015, spent last season on loan with the Italian champions.
The winger impressed during his 40 appearances and ended the campaign with five goals to his name.
It was thought that Cuadrado would join Juventus permanently after that successful season, but so far new Chelsea boss Antonio Conte has dug his heels in.
The Italian believes the 28-year-old could have a part to play over the course of the year and he is reluctant to sell him.
However, according to Gazzetta dello Sport, that may change as Cuadrado has now directly told Chelsea he wants to leave.
The Colombia international has enjoyed the best days of his career in Italy and he is eager to return.