Russian outlet Soviet Sport claims that members of the Reds’ scouting department were in attendance as Azmoun’s Zenit St Petersburg faced Lokomotiv Moscow at the VTB Arena in Moscow.
Reigning champion Zenit was defeated 3-2 but Azmoun scored two goals either side of half-time.
The 24-year-old only joined the Gazprom-owned outfit back in February for a fee of around £10.5 million, having neared a loan move to Wolves in January which fell through on deadline day.
Iran international Azmoun, formerly nicknamed ‘The Iranian Messi’, has long been tipped for a move to Europe with Liverpool credited with interest in the player since 2016.
And Soviet Sport say that Liverpool scouts left Russia with positive reports of Azmoun’s display against Lokomotiv.
The ex-Rubin Kazan front-man has enjoyed a positive spell in his new country so far, with 14 goals and three assists in just 17 appearances across all competitions.
Azmoun insists he rejected the chance to join up with Jurgen Klopp in January 2017.