The Yellow Canaries are looking to bolster their attacking options in midfield after failing to win the league title last season.
Fenerbahce did however, finish runners-up and will compete in the Champions League next season.
Malli was born in Germany but is also a Turkish citizen and plays for the Turkey national team.
The attacking midfielder had an impressive season at Mainz scoring 11 goals in 34 games and will accompany Turkey in France this summer for the European Championships.
The Istanbul giants could however, face competition from Tottenham who have been keeping close tabs on Malli all season.
Spurs have been heavily linked with Malli in recent months and had scouts present at Maniz’s game against Hanover 96 recently. Sport Bild have also reported that Tottenham are interested in signing the Turkish international who has had an excellent season so far.
Malli still has another two-years on his contract and Mainz reportedly want £8 million for their star man.