According to reports in France, a deal to make Henry the successor of former boss Gus Poyet has now fallen through as Bordeaux look towards former Leicester and Chelsea manager Claudio Ranieri.
However, 'RMC Sport' report that talks could be reinitiated, despite coming to an end following Henry's significant salarial and logistical demands.
Nicolas de Tavernost, Bordeaux's owner, insisted all options are "still open", saying: "No decision until next week."
"We are working day and night to have the best coach possible. It's not easy."
Henry, 41, was the assistant manager of Belgium's national team, who reached the World Cup semi-finals this year.