Henry, who enjoyed a stellar club career with Arsenal and Barcelona, had been assistant coach for the Belgian national team.
He replaces former Uruguay international Gus Poyet at the helm of the Ligue 1 side. Poyet had been suspended last week for his comments on the club's transfer dealings after the sale of striker Gaetan Laborde to Montpellier.
RMC Sport said the club's prospective new American owners, who agreed to buy Bordeaux from M6 at the end of July, still needed to validate the agreement.
Henry, France's record goalscorer, would join Patrick Vieira as the latest member of the national team's 1998 World Cup winning squad to manage in the country's top flight after he joined Nice this season.
Bordeaux has won the French league title six times with the last championship coming in 2009 under the leadership of another 1998 World Cup winner, Laurent Blanc.
(Reporting by John Irish; editing by John Stonestreet and Christian Radnedge)
