The Turkish Süper Lig broadcaster report that Fenerbahçe sporting director Giulliani Terraneo recently returned from England having struck a deal with Manchester United.
Terraneo also held talks with Nani’s agent Perrano in Istanbul last week and agreed personal terms with Nani.
The Sun believe that the Portuguese international will sign for a €7 million transfer fee on a four year deal worth £3 million per-season.
The 28-year-old still has three-years left on his contract but is expected to leave the Premier League giants over the summer after falling out of favour with United boss Louis van Gaal.
Nani spent last season on loan at Portuguese giants Sporting Lisbon where he scored 12 times and provided eight assists in 38 games in all competitions.
The Yellow Canaries are planning to complete all their transfers before the start of the pre-season and have already signed Denmark international Simon Kjaer on a four year deal.