The report claims that the Yellow Canaries have made the Blues a bid and that negotiations are ongoing.
The Istanbul giants have been closely linked with a move for Remy over the past season and reportedly made a loan offer in January.
Rey faces an uncertain future at Chelsea following the appointment of Conte as head coach and remains to be seen if the current Italian national team boss has plans for the French striker next season
The powerful striker lost his place in the France national side after struggling to break into the Chelsea side last season.
Remy provided one assist and scored three goals in 20 appearances for the London based club this season.
The 28-year-old joined joined the Blues on a £10.5m move from QPR last year, has struggled with injuries and a lack of form in recent months but still has another two-years remaining on his contract.
Fenerbahce finished last season as runners-up and will play Champions League football next season.