ESPN previously reported that the 34-year-old had been keen on a move to Italy after turning down a 12-month contract extension with the London club in June.
On Friday, Giroud thanked Chelsea's fans, players and coaches for "special moments" as his switch away from the Premier League club drew closer, with British media reporting Italian side AC Milan would be his next team.
"To all the Blues, to my team mates, to all my coaches, to the whole club, a huge thank you for these special moments," Giroud said on Twitter. "I'm starting a new journey with a light and happy heart. Our victories in the FA Cup, Europa League and Champions League have been magnificent."
The France international moved to Chelsea from Arsenal in January 2018 and scored 39 goals in 119 appearances for the club. Giroud also won an FA Cup, Europa League and Champions League in his three years at Stamford Bridge.
Giroud was a World Cup winner with France in 2018 and part of the side which was knocked out of Euro 2020 at the round-of-16 stage by Switzerland on penalties.
Milan, who finished second in Serie A last season, begin their 2021-22 league campaign away to Sampdoria on Aug. 22.