According to Sky sources, Henry has 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 (GACP).
The Ligue 1 side recently suspended manager Gus Poyet after he labelled the club “a disgrace” following the sale of striker Gaetan Laborde to Montpellier without his knowledge.
The club also sold star man Malcom to Barcelona for £36m over the summer, and Henry is understood to have misgivings over guaranteed future investments in the first-team squad.
Importantly, neither Henry nor the club has confirmed that negotiations have broken down, but at this stage it appears unlikely that his first job in management will be at Bordeaux.
Henry, who left his role as a Sky Sports pundit in the summer to concentrate on his coaching career, was linked with the Aston Villa job earlier this summer.
The 41-year-old has been working as an assistant to Belgium boss Roberto Martinez since 2016 and was part of the backroom staff that led the team to third place at the World Cup in Russia.