The report states that Turan has been placed on the transfer list after making just nine starts for the La Liga champions.
The Sun go onto state that Barcelona are open to offers for Turan over the summer transfer window.
The Turkish newspaper joined Barcelona on a five-year deal but had to sit out the first six-months on the sidelines due to the FIFA transfer ban that was imposed on the club.
The 29-year-old ended the season with a two goals and three assists in 810 minutes of La Liga football.
Turan was however, given just 46 minutes of playing time by head coach Luis Enrique in his last four league games.
Despite being linked with a move away from Barcelona boss Luis Enrique has always maintained that Turan will be at the club next season.