The free-scoring French international has been linked with Arsenal, but the Gunners face stiff competition for his signature from a host of top European clubs.
However, Marseille sporting director Jose Anigo has tipped the prolific forward to remain at Montpellier.
He told French Television station OMTV: "This is a player who wants to play in the Champions League next season and it seems we are not going to get there.
"That is why I do not believe we have any chance to sign him. I don't believe he will leave France this summer because he will want to play with Montpellier in the Champions League."
Giroud is the leading goalscorer in Ligue 1 this season, and found himself among the goals in this evening's 3-1 win at Marseille.