Azmoun has scored 16 goals in 22 appearances for his country and has been watched by a number of top clubs in Europe.
The 21-year-old was linked with a move to Everton last season.
Liverpool boss Jurgen Klopp admitted last week he might need to strengthen in January following a season ending injury to Danny Ings.
Azmoun joined Rostov on a permanent deal in the summer in controversial circumstances.
After spending a period on loan at Rostov, they triggered a buy-out clause which Kazan claimed was non-existent.
The case went to the Court of Arbitration for Sport, which ruled in favour of Azmoun and Rostov.