According to L'Equipe (via The Mirror), representatives from the two clubs met in London this week.
Koscielny infamously decided against joining the Arsenal squad on their pre-season tour of the US.
The 33-year-old felt he was wronged by the club after they rejected his request to be allowed to leave for free.
Arsenal, however, appear to have little sympathy for the player who has one year left on his contract and has now been forced to train with the under-23s.
Bordeaux will reportedly try to persuade Arsenal to release Koscielny, but Arsenal are not expected to budge from their £9m valuation.
The Ligue 1 outfit's president Eduardo Macia has spoken publicly about wanting Koscielny.
"If there's a chance, we would be thrilled to have him [Koscielny] come here.
"He's a leader, an intelligent defender and one of the best defenders whom I've seen in my career.
"It would be incredible to have a player of his quality at Bordeaux. But he has a contract at Arsenal."
The Premier League transfer window closes on August 8, but English clubs can still sell players after the deadline.
However, Arsenal are reportedly seeking to sign a centre-back themselves, and if they fail to do so, they may force Koscielny to stay and see out his contract.
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Quiz! Can you name all 27 Community Shield goalscorers from the past decade?
