The report also claims that the Yellow Canaries will put forward a three-year contract with a clause for a one-year extension worth £3 million per-season.
NTV Spor claim that an agreement is almost in place with Nani after his agent Federico Pastorello travelled to Istanbul and held talks directly with Fenerbahçe sporting director Giuliano Terraneo at the club headquarters.
The Istanbul giants have had a busy summer transfer window and already signed Lille defender Simon Kjaer on a four-year deal last week.
Fenerbahçe have one of the oldest squads in the league (30) and want to lower their average age by signing new players under the age of 30.
The Portuguese international spent last season on loan at Sporting Lisbon scoring 12 goals and providing eight assists in 38 games.
The Istanbul giants are coached by Portuguese manager Vitor Pereira who joined the club a fortnight ago.