The agent of Nani, Federico Pastorello arrived in Turkey yesterday to enter negotiations with Fenerbahçe according to NTV Spor.
Nani has been a Manchester United player since 2007 and has won four league titles as well as the Premier League, Champions League and World Club Cup.
The 28-year-old winger fell out of favour with former manager David Moyes and his replacement Louis van Gaal before joining Sporting Lisbon on loan last season.
The Portuguese international scored seven times in 27 appearances for Sporting last season.
Fenerbahçe are coached by Portuguese manager Vitor Pereira and Nani would have two Selecao teammates were he to join – Raul Meireles and Bruno Alves.
The Yellow Canaries finished last season as runners up and are expected to invest heavily over the summer having booked a place in the Champions League for the first time in two years.