The Iranian star is said to have had his head turned by the alleged interest from clubs including Everton and Celtic.
Celtic were reportedly keen to sign Sardar Azmoun over the summer.
Celtic may have got a lucky escape deciding against the signing of Sardar Azmoun from Russian Premier League side Rubin Kazan in the summer.
Celtic are keen on Sardar Azmoun and he plans to decide his future next year.
Celtic were reportedly considering making an approach for Sardar Azmoun - who would have cost them £10m.
Celtic want a new striker to compete with Leigh Griffiths and Moussa Dembele, with FC Rostov's Sardar Azmoun in the frame.
Celtic boss Brendan Rodgers originally was linked with Sardar Azmoun in the early part of the transfer window.
Celtic were reported to be eyeing Sardar Azmoun if Moussa Dembele leaves.
Sardar Azmoun scored against both Bayern Munich and Atletico Madrid in the Champions League and he has allegedly piqued Celtic's interest.
Celtic may be able to sign Sardar Azmoun despite his Liverpool snub.