The Rossoneri are looking to reinforce their midfield line this summer with the departure of Andrea Pirlo to Juventus and the uncertainty surrounding Gennaro Gattuso's future at the club.
The 24-year-old Fabregas is considered among the reported targets of Milan.
"From here until August 31, Milan will buy a player that will be a left midfielder," Galliani said to Gazzetta dello Sport.
"(Napoli winger Marek) Hamsik and Fabregas can play in that role.
"We have never asked for Hamsik although we don't rule him out and as for Fabregas, he is too expensive."
Arsenal are understood to value Fabregas at £60million.
The Spain international is under contract with the London outfit until June 2014.