Poyet was suspended last Friday for a week after an angry outburst at club bosses over their transfer policy.
Though the Uruguayan remains under contract, he has effectively been pushed out of his role, creating a void that Bordeaux are keen to fill with a high-profile name.
Thierry Henry has taken his first tentative steps into the world of management by accepting the position at Ligue 1 side Bordeaux.
Wanting to concentrate on his managerial career, Henry quit his role as a Sky Sports pundit after the World Cup, where he was assistant manager to Roberto Martinez as Belgium finished third.
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This is always the way, but it is clear that defensively we suffered three goals but also we had chances to do more goals. But also they had options, chances to score and effectiveness is the difference.
Primarily, Henry wished to know about the funding he will have to strengthen the squad when GACP complete their takeover and whether he'd have flexibility in the market.
Wenger ended his 22-year tenure at Arsenal in May.
Telegraph Sport understands that there is interest in Henry from Bordeaux, a national association and at least one club in England.
Bordeaux have lost their first two games of the season, a 2-0 loss against Strasbourg and a 2-1 defeat away to Toulouse, to leave them without a point at the start of the season and within the quagmire of the relegation zone.