This is according to Sky Sports, who claim the Ligue 1 club are willing to pay £20million for the 30-year-old striker.
The report adds Marseille believe they can convince the forward to return to his homeland.
Giroud has struggled to hold down a regular place at Arsenal first team this season.
The France international has had to compete with Alexis Sanchez and Danny Welbeck for the sole striker spot in Arsene Wenger’s side.
Giroud has only made eight Premier League starts this term, and he admitted last month he has found being on the sidelines this season difficult to cope with.
“For me, the most difficult moment was after scoring an important goal against Manchester United, I thought I had scored some points and would start against PSG… but no,” Giroud told French magazine L’Equipe.
“So I asked to speak to the coach to understand. I accept his decisions but I tell him what I think, without being negative, without antagonising.
“There are no misunderstandings between us.
“The coach explained to me that he was adopting this system [4-2-3-1] to find a defensive balance.
“Sometimes, I played with Sanchez on the left, but the rest of time I have to adapt to the tactical decisions.
“I understand the notion of balance once the ball is lost.”
Giroud, who originally joined Arsenal in 2012 from Montpellier, has nine league goals this term.
And he is under contract with the Gunners until the summer of 2020, after he signed a contract extension back in January.
