The out-of-favour Barcelona midfielder is seeking a move away from the Nou Camp, having fallen well down the pecking order.
Rumours have surfaced that Barca head coach Ernesto Valverde is unhappy with the Turkish international and is ready to offload him in January.
Manchester United boss Jose Mourinho reportedly held talks with Turan over the summer in a bid to convince him to join the Old Trafford club.
However, Turan remained in La Liga and now, according to Turkish outlet Mynet, United will face competition for his signature from Fenerbahce.
Both clubs are eyeing January moves and it remains to be seen whether Turan favours a switch to the Premier League.