Reports from China suggest Drogba, 34, is seeking close to £8.5 million for an 18-month deal with the former European champions if he terminates his contract with Shanghai.
Drogba is currently playing at the African Cup of Nations with Ivory Coast, but is apparently in the market for a move after just months in China.
AC Milan coach Massimiliano Allegri said: "If Drogba joins Milan then I'll obviously be happy - just like I’ll be happy if any champion joins us and strengthens the squad. I think that if Galliani gets the chance to sign one or two champions, then he will do it.
"A player like Drogba would certainly improve both the technical level and self-esteem of the team. We'll see what the club are able to do, otherwise we'll carry on as we are, looking to have a good second half of the season because that is our objective.
"It will be difficult, though, because we still have to face some teams with a high level of technical quality."