Massimiliano Allegri has been pushing his side to sign a new defender in the remaining days of the transfer window following the sale of Malick Thiaw to Newcastle.
Now, according to Fabrizio Romano, the Italian giants have made a move to sign Akanji, 30, from Premier League side Man City.
Milan are yet to make an offer for the Switzerland international but have made contact with City in order to discover potential terms of a deal.
Turkish side Galatasaray were previously linked with a move for Akanji earlier in the summer but that has since fallen through.
Man City are understood to be open to selling the central defender with the decision on whether to leave or not now up to Akanji.