The Ligue 1 side have concluded that Debuchy is too expensive.
Debuchy is seen as too expensive by Bordeaux
Debuchy looks unlikely to be a part of Arsene Wenger's first team plans next season, with Hector Bellerin set to continue as his preferred choice, meaning they are willing to listen to any offers.
The former Newcastle United star spent the second half of last term on loan at Bordeaux, although injuries restricted him to just nine appearances.
That run of form saw the defender overlooked by France boss Didier Deschamps at the recent European Championships, although Sky Sports claimed Bordeaux were still keen on a deal for the 30-year-old.
However, Sud Ouest have now claimed that Bordeaux have cooled their interest, concluding that even if they could reduce the Frenchman's £70,000-a-week wage demands, a deal would still be too expensive.
It remains unclear as to whether Debuchy, who still has three years left on his current deal, would be willing to stay at the Emirates, having played just 12 games since making the move from Tyneside in July 2014.