The Brazil international returned to his homeland in 2011 after spending seven years with Porto and Sevilla.
He is now 32 years of age, but remains a regular source of goals at club level and an important part of national coach Luiz Felipe Scolari's plans leading up to the 2014 World Cup.
Fabiano has now revealed that he is intrigued by the prospect of heading to Russia to link up with Anzhi - a side who employ Brazilian legend Roberto Carlos as their sporting director and have Cameroonian striker Samuel Eto'o on their playing staff.
He said: "I can't say anything concrete about it, but anything can happen.
"It may be interesting to play for Anzhi."
Fabiano has also offered his opinion on international strike partner Neymar and the ongoing rumours linking him with an imminent move to Spanish giants Barcelona.
He added: "I think the Spanish league would fit Neymar's style very much, as they play very technical football.
"I don't know if it would be good for him to join Barcelona because they have Leo Messi, but I'm sure they could play in the same team.
"It wouldn't be easy for Neymar in the beginning as he doesn't have experience at the highest level."