The Turkish newspaper Fanatik has reported that the Toffies are among a handful of top-flight teams racing to sign the 29-year-old.
The report states that the Portugal international wants to return to the Premier League just a year after joining from Manchester United.
The daily sports paper also claims that Nani included an option in his contract giving clubs from England, Spain and Italy a minimum fee release clause of £6.4 million.
However, the option is not available for clubs outside the three countries mentioned above which explains how Fenerbahce were able to turn down a €20 million bid from China over the winter transfer window.
Nani joined the Yellow Canaries on a three-year deal from United and has hit the ground running at his new club scoring 10 goals and providing eight assists in all competitions for the Istanbul giants this season.