He's scored one goal for the Blues, but his lack of opportunities has left him frustrated.
The former Arsenal striker is seeking game-time to enhance his chances of playing a major role for France at next summer's European Championships.
The French national team coach Didier Deschamps has been openly critical of Frank Lampard's omission of Giroud.
“You know, Olivier suffers his situation. Before January, nothing will move,” Deschamps said in October.
“After, it will be up to him to see. I hope he will have more playing time.
“But we cannot take away his goals, his good performances. He scores, he makes others play well."
Antonio Conte is keen to add to his squad, which he feels is too lightweight to compete effectively in both the Serie A and the Champions League.
Conte was openly critical of Inter's recruitment after their loss to Borussia Dortmund in the Champions League in midweek.
''We have too limited a squad to face both Serie A and Champions League football this season," he told press.
''I mean in terms limited in terms of numbers and quality.''
A caveat for Giroud is that he will have to compete with both Romelu Lukaku and Luataro Martinez - both of whom have been in good form.
Lukaku joined last summer for £72m from Manchester United and was joined on loan by Alexis Sanchez.
The 33-year-old may fear he will find himself in a similar rut in Milan, with two younger strikers ahead of him in the pecking order.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.