The Yellow Canaries are reported to have put forward a three year deal with an option to extend to four seasons.
The 28-year-old winger would become the most highly paid player for Fenerbahçe if the transfer is finalised.
Nani is expected to arrive in Turkey this week to complete his Fenerbahçe transfer. The Yellow Canaries would have completed their second signing of the transfer window should the deal go ahead. The Istanbul giants also signed Lille defender Simon Kjaer on a four year deal last week.
Fenerbahçe plan to complete their transfers before the start of pre-season training and the Champions League play-off round.
The Yellow Canaries will try to book a place in the Champions League group stages via the playoff route after finishing the league as runners up last season.