The transfer sees Contento's 19-year association with Bayern come to an end after he joined the club at the age of five in 1995.
Contento, a left-back, made his debut for the first-team in 2010, but was never able to establish himself as a regular.
He made 69 appearances for the club, but will now hope to gain more frequent action in France.
The Bavarian club revealed that they retain 50 per cent of the player's economic rights.
Should he pass his medical at Bordeaux, he will join up with another former Bayern full-back in Willy Sagnol, who is now Bordeaux's head coach.
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