Robben has been closely linked with a move for the Netherlands international and the Milliyet claimed that the Yellow Canaries had made a bid for the experienced winger.
There was further speculation that a transfer could be on the cards following Robben not joining the rest of the Bayern squad on the China tour.
The Yellow Canaries did however, put forward a €6 million offer for the 31-year-old winger last season according to the Akşam newspaper.
Fenerbahçe are expected to make another signing should they progress past Shakhtar Donetsk in the Champions League preliminary play-off tie.
The Istanbul giants have already made eight signings this season including Robben’s Netherland international teammate Robin van Persie from Manchester United.
Fenerbahçe also signed fellow United player Nani on a three-year deal. Lille defender Simon Kjaer, last seasons top scorer Fernandao and Brazil international Josef De Souza also joined the Yellow Canaries over the summer.