According to Sky Sports News, Henry told the club on Monday that he didn't want the job, having asked for the weekend to consider the post.
"According to Sky sources, Henry had concerns over the availability of funds and a lack of clarity over the ownership of the Ligue 1 club—which is presently in takeover negotiations with Miami-based investment fund General American Capital Partners," it's added.
Former Leicester City boss Claudio Ranieri is now said to be Bordeaux's top target to fill the position following the suspension of Gus Poyet.
According to French football journalist Jeremy Smith, Bordeaux also had some concerns about Henry:
Jeremy Smith @jeremysmith98
As for Bordeaux, they had already been turned down by Laurent Blanc and Remi Garde. Ranieri has been mooted but he's thought not keen to return to another job in France. Other names mentioned - ex-coach Ricardo, Herve Renard, Valid Halilhodzic - not generating too much enthusiasm
As noted in the report, Poyet was suspended by Bordeaux after labelling their owners a "disgrace." That was after the club sold forward Gaetan Laborde to Montpellier without his permission.
Henry was part of Roberto Martinez's coaching staff with the Belgium squad during the FIFA World Cup. The team enjoyed a successful tournament, as they finished in third place.
However, Henry now appears keen to get into management and left his role as a pundit with Sky Sports this summer to further focus on his coaching.
French football journalist Andrew Gibney believes a move to Bordeaux would've been an ideal first step into management for Henry:
If the 41-year-old enjoys a coaching stint anywhere near as successful as his playing career he'll be a tremendous success.
Henry is an icon at Arsenal, as he's the club's all-time top goalscorer and was the driving force behind two Premier League title triumphs. The Frenchman moved to Barcelona later in his career, where he won La Liga twice as well as the UEFA Champions League.