The Gabon international is under contract at Les Verts until 2016, but it is thought that a €15million fee could persuade St Etienne to sell.
The striker is also thought to have attracted interest from Arsenal, Tottenham, Newcastle and Fiorentina.
French magazine L'Equipe has revealed that Cameroonian Eto'o has recommended to Anzhi to offer Aubameyang a lucrative contract, quoting the striker as saying: "Such a proposition obviously makes me think about it."
Meanwhile, St Etienne manager Christophe Galtier has not given up hope of keeping the forward at the club.
"[Aubameyang]'s under contract," Galtier told RMC
"He has been given permission to leave, but there is a value attached for that permission to be granted. If I can keep him, then of course I would like to."
Meanwhile, Aubameyang's brother, Willy, who is also at St Etienne, admits that his brother will make his own decision on his future.
"I am looking forward to next season with or without Pierre," Willy said.
"He is charge of his own future. It is his decision to what we wants to do."