La Gazzetta dello Sport have reported that Milan will pay Chelsea €1 million in compensation to sign the player, who had his contract automatically renewed by the Blues this summer as it was set to end.
Giroud is expected to arrive in Milan on Thursday ahead of undergoing his medicals on Friday.
The forward will sign a two-year deal with the Serie A club worth €3m per year. There will also be an option in place that will allow the club to add a third year onto his stay in Lombardia.
Giroud will then report to training after his post-Euro 2020 holidays after July 26.