The Moroccan international, who has netted 15 times for his country, has been linked with a move to the Premier League throughout the close-season.
His 13 goals last term helped secure a first league title in 10 years for les Girondins, but Chamakh has made no secret of his desire to leave the Stade Chaban Delmas.
He has previously claimed to have agreed terms with Sunderland, Fulham, Blackburn Rovers and Tottenham Hotspur, while Arsenal are now believed to have withdrawn their interest in the striker after refusing to up their offer of £5 million.
Hammers boss Gianfranco Zola has already been linked with several high-profile forwards this summer, including Barcelona's Eidur Gudjohnsen, Inter Milan's Mario Balotelli and Bayern Munich talisman Luca Toni, as he bids to bolster his front-line.
But, according to L'Equipe, the Italian's opening offer for Chamakh has been turned down by the French champions.
While Bordeaux president Jean-Louis Triaud has indicated that the striker may not be interested in joining the Upton Park outfit, with the Moroccan having had his heart set on a switch to the Emirates Stadium.
"That offer was not acceptable. So it was no," he said.
"Moreover it is not obvious the player wants to join West Ham."
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