According to Sky Sport Italia, the 22-year-old is under contract until June 2018 and has been in Russia since January 2013 for Rubin Kazan and then Rostov.
The Italian team wants to sign Azmoun as a replacement for Fabio Borini.
Borini is going to join Milan.
Azmoun scored 12 goals with three assists in 41 competitive games for Rostov this season, while he has 19 goals in 26 senior caps for Iran.
The Iranian forward had been already linked with Liverpool, Celtic and Sunderland.