Nani is currently available on an €8.5 million release clause which is open to clubs from the Premier League, La Liga and Serie A.
Clubs wishing to activate his release clause will however, have until the end of June to do so according to the Aksam newspaper.
Nani joined the Yellow Canaries on a three-year deal from Manchester United last season but the Red Devils have also been closely linked with their former winger.
United are reported to have agreed personal terms with Nani according to Turkish media outlet DHA but are reluctant to match his release clause as they would end up paying more than they sold him for last season.
The report claims that Fenerbahce will be able to name their price after June and considering they value the Portuguese international as being worth €15-20 million clubs wanting to sign Nani with immediate effect would be getting a bargain.
Nani is however, unlikely to make a decision on his future until after Euro 2016 which will end in mid-July.