That role saw the 1998 World Cup winner return to a global showpiece over the summer in Russia.
He helped Belgium to a third-place finish before deciding to quit his punditry posts for 2018-19 in order to focus on coaching.
The 40-year-old is eager to get into management and could be offered a first post back in his homeland having previously been linked with Aston Villa during pre-season.
Bordeaux are set to find themselves with a void to fill after suspending Poyet in the wake of comments from the Uruguayan criticising the club.
Henry is intrigued by the role and has asked questions about it.
It is understood that he is the preferred option for incoming owners General American Capital Partners (GACP).
American businessman Joseph DaGrosa and his GACP partners are set to take control of the Ligue 1 outfit in September.
DaGrosa has floated the idea of bringing the former striker on board, with Bordeaux now weighing up their options ahead of a trip to Toulouse on Sunday.