The 28-year-old, joined Fulham on loan from Stuttgart in January, and has scored five goals in his first four games.
The Russian's contract will expire at the end of the season, however, and Villa boss Alex McLeish has hinted that he could make his move.
"Fulham did well in January and brought Pavel Pogrebnyak into the setup. He is a player who I've seen and have been in touch with his agent over the past few years trying to sign him, but the timing was just not right," McLeish told The Mirror.
"I need to be careful what I'm saying ... But we are lacking experience, and hopefully that is something we can put right."
It is understood that Fulham have first refusal on a permanent deal.