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, with Arsenal - his preferred destination - seeing a £5 million bid turned down by Bordeaux.
Gunners boss Arsene Wenger previously insisted he would not be increasing his offer nor his efforts to sign the striker, stating he would rather wait a year and land the front-man for free.
But despite suggestions that the Emirates Stadium were planning to submit a second bid for the forward, the Daily Mirror have now claimed that Wenger has turned his back on a deal to bring him to North London.
Arsenal's exit leaves fellow London rivals West Ham and Fulham to fight over Chamakh's signature.
The Hammers are desperate to sign a forward to support Carlton Cole at Upton Park, with Barcelona's Eidur Gudjohnsen, Inter Milan's Mario Balotelli and Luca Toni of Bayern Munich also being linked with Gianfranco Zola's side.
While the Cottagers are thought to be willing to offer Bordeaux French forward Diomansy Kamara in part-exchange for Chamakh's services.
NEWS: Fulham eye Chamakh swap swoop
NEWS: Hammers set for late Chamakh swoop
NEWS: Triaud - No new Chamakh offer from Arsenal
NEWS: Chamakh losing sleep over Arsenal ordeal
NEWS: Blanc questions Arsene interest in Chamakh
NEWS: Wenger - Chamakh asking price still too high
