The 34-year-old striker currently finds himself as first-choice at Stamford Bridge. However, the France international will be out of contract in the summer, which inevitably leaves him being linked with a move elsewhere.
Marseille manager Andre Villas-Boas wants to sign a new striker next month and Giroud is one of their primary targets.
The aforementioned report suggests Andre Villas-Boas’s side are ready to make an offer for the Frenchman during the mid-season transfer window.
Marseille are currently the No.1 favourites for the Ligue 1 title, sitting five points adrift of league leaders Lille with two matches in hand.
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Giroud in search of regular football ahead of next summer’s rescheduled Euro 2020 and this is likely to influence his future when the mid-season transfer window reopens in January.