The Yellow Canaries newly appointed manager Vitor Pereira has instructed the club to open negotiations for the Portuguese international.
The Daily Star claim that United have slapped a €14 million price tag on Nani and that he has no future at Old Trafford. Inter Milan are also said to be keen on the 28-year-old.
Nani currently earns €140,000 a week at Manchester United which could prove to be a stumbling block for Fenerbahce. The Portuguese international has three years remaining on his contract with United.
Fenerbahçe finished last season as runners up and were unable to prevent rivals Galatasaray winning the league for a record 20th time and earning the right to wear the fourth star.
The Yellow Canaries did however, book a place in the Champions League play-off stages and will enter European competition for the first time in two years.