The striker is a free agent, as his contract with Chelsea expires on June 30, and he is considering various different proposals.
According to Calciomercato.com, those include offers from West Ham and also MLS club NYCFC.
However, Giroud believes he can still make the difference at the top level and is intrigued by the opportunity to play in Serie A and the Champions League for Milan.
A basic agreement has been reached with the Rossoneri for a two-year contract at €4m per season plus performance-related bonuses.
Sky Sport Italia, le10sport.com and others are all confident Milan will get the deal done.
The final details should be resolved next week, as the France international wants the situation resolved ahead of Euro 2020.