Chelsea triggered an option last month to extend his contract by a further 12 months but the 34-year-old wants more regular game-time and is keen on a move to Italy.
Giroud fell down the pecking order at Stamford Bridge last season following the arrival of Timo Werner from RB Leipzig with Kai Havertz also preferred at times as a false nine by Chelsea boss Thomas Tuchel.
The club are in the market to sign a forward this summer and are expected to try and prise Erling Haaland away from Borussia Dortmund. They are also keeping tabs on Harry Kane's future at Tottenham Hotspur among other possible targets.
Chelsea are willing to part with Giroud, who moved across London from Arsenal in January 2018 and has scored 39 goals in 119 appearances.