Chantome has risen through the youth ranks at PSG and his departure brings an end to his 15-year association with the captial club.
The 27-year-old has won every major domestic honour with the Parisian giants and made 28 appearances in their 2012-13 title-winning campaign.
Chantome has slipped down the pecking order under Laurent Blanc and spent last season on loan at Toulouse.
Bordeaux, who sit seventh in Ligue 1, have now given Chantome the possibility of playing regular first-team football.
A statement from PSG read: "The entire Paris Saint-Germain family would like to thank Clement for his commitment and professionnalism over the years and wish him every success for his new sporting project."
Bordeaux also revealed on Friday that midfielder Clement Badin has gone out on loan to Spanish third tier side Real Aviles.
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