Bordeaux tweeted a picture of Contento signing his contract before posting a story on their website confirming the transfer from Bayern.
The transfer was completed soon after Bordeaux confirmed they had sold another left-back - Lucas Orban - to Valencia.
Contento passed a medical on Monday and will be officially presented to the media on Tuesday but Bordeaux coach Willy Sagnol had already confirmed on Saturday that they were close to bringing the German to France.
"As soon as we started talking about bringing in a left-back, we knew that Contento could be an interesting opportunity," Sagnol said.
"He spent five years with the Munich first-team squad, even if he didn't play all that often. He's played in the Champions League against French clubs. He's a highly motivated player who'll be out to prove what he's capable of."
Contento played just 49 Bundesliga matches with Bayern despite having been part of the senior squad at the Allianz Arena since the 2009-10 season.
The 24-year-old first signed for Bayern at the age of five.
Orban leaves Bordeaux after just one season with the Ligue 1 club where he played 27 league matches.
