The Turkish sports newspaper claims that the Black Eagles want to loan the Belgium international until the end of the season.
Fellaini scored seven times in 31 appearances for the Red Devils last season however, TalkSport report that the 29-year-old could leave United over the transfer window.
Speaking to Talksport Warren Haughton said Galatasaray and Fenerbahce are both interested in the powerful midfielder.
Besiktas are however, most in need of a midfielder out of the three clubs in contention for the Belgium international.
The Black Eagles were closely linked with Switzerland international Gokhan Inler before his £5 million move to Leicester City.
The Istanbul giants will be without central midfielders Veli Kavlak and Tolgay Arslan until the turn of the year due to injuries.