With Thomas Tuchel making a top-class centre forward a priority this summer, Giroud is set to be pushed further down the pecking order at Stamford Bridge.
Milan are believed to be confident they can convince the 34-year-old to switch to Serie A on a two- or three-year deal.
Chelsea have been working on a list of potential strike targets for several months, with Erling Haaland their long-term top choice, while Romelu Lukaku, Sergio Aguero and Harry Kane were also included.
Lukaku was seen as the most attainable option, but the Belgian in intent on remaining at Inter Milan. Aguero has signed for Barcelona on a free transfer and Kane is believed to favour a move to Manchester City.
Chelsea are continuing to explore the possibility of signing Haaland – but Borussia Dortmund are adamant he will stay for one more season. The Germans will also be under less pressure to sell, with Jadon Sancho set to join Manchester United.
Tuchel wants to shake up his attack, with Timo Werner enduring a difficult first season in English football and Tammy Abraham failing to convince the German he can lead Chelsea’s line.
Tuchel would be happy to keep Giroud as a squad player, but the veteran’s desire to play more regularly is expected to see him leave Stamford Bridge after three-and-a-half years.
It is unclear if Giroud will be allowed to leave as a free agent after having his contract extended, but Milan do not want to pay a fee.
The Italian club are also set to trigger an option to buy on-loan Chelsea centre-back Fikayo Tomori.