The Russia international has been in fine form since joining the Cottagers in January, netting five goals in four appearances, including an impressive hat-trick against Wolves.
McLeish has long been an admirer of the Russian and, after failing to land him while at the helm of Birmingham, would like to finally get his man.
The Scot told The Daily Mirror: “Fulham did well in January and brought Pavel Pogrebnyak into the set-up.
“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 quite right.
"But we are lacking a bit of experience, and hopefully that is something we can put right.”
Aston Villa are 11/2 with Bet365 to beat Bolton 1-0 in their next Premier League encounter.
The Villa Park chief was cautious in announcing any potential deal for Pogrebnyak, whose short-term contract with the Cottagers expires at the end of the season.
“I don’t want to be in the bad books with Fulham talking about their player,” he added.
"He’s a player I flirted with in the past, but I shouldn’t really be talking about someone else’s player.
“I need to be careful what I’m saying.”
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