The Yellow Canaries have been closely linked with the Netherlands international over the past few seasons.
Netherlands international teammates Dirk Kuyt and Wesley Sneijder are reported to have encouraged Van Perise into considering a move to Turkey.
Van Persie has been blighted by injuries this season but has still managed to hit the back of the net 10 times in the Premier League.
The 31-year-old has 15 months on his contract but is rumoured to be considering a lucrative long-term contract elsewhere as United have a one-year contract policy for players over the age of 30.
Van Perise has been linked with Fenerbahçe in the past and Kuyt has claimed in the past that he is fond of the club.
“Van Persie knows the club well, I have told him all about Fenerbahçe and he has told me that he would like to play in Turkey one day,” Kuyt told Lig TV in an interview in 2012.
Fenerbahçe are planning to restructure their squad next season if they secure a Champions League place at the end of the season.