The report claims that Juve are planning to strengthen their squad should Manchester United complete the £100 million of Paul Pogba.
Turan has hinted in the past that he does want to play in the Premier League one day and was reported to be looking for a flat in London last year.
“I am planning to buy a place in London, I want to have a place there at all times, you never know when it could come in useful, if I have kids they could study there,” Turan told Hürriyet in an interview.
The Sun also claim that Barcelona could be willing to sell him for a cut-price £23 million deal despite signing him for more last season.
Despite being linked with a move aboard Turan does have another four-years left on his contract.
Turan missed the first half of last season due to the FIFA transfer ban imposed on the club but he did make 25 appearances after making his debut in January and was part of the La Liga and Spanish Cup winning sides.