The France striker arrived at Milan’s La Madonnina clinic at approximately 7am on Friday morning ahead of finalising his move from Chelsea.
The transfer fee for Giroud is understood to be around €1m (£855,000), with an extra €1m (£855,000) in bonuses subject to a number of appearances and qualification for the Champions League.
The 34-year-old signed a new one-year deal at Chelsea last summer and the club took up an option to extend it by a further 12 months in June.
Arsenal sold Giroud to Chelsea for £18m in January 2018, six years after signing him from the Ligue 1 title-winning Montpellier side for £10.8m.
He has scored 39 goals in 119 appearances during his time at Stamford Bridge, winning three major trophies, including the Champions League title in May.
Giroud was involved in France’s disappointing Euro 2020 campaign as they were knocked out by Switzerland on penalties at the round-of-16 stage following a 3-3 draw after extra-time.
AC Milan have already signed Fikayo Tomori from Chelsea this summer, as they activated their £25m option to buy the young defender following a successful six-month loan spell, and are interested in Hakim Ziyech and Tiemoue Bakayoko.
The Italian club finished second in Serie A last season, securing a return to the Champions League following a seven-year absence.
