The former Manchester United defender was not given an extension to his contract pre-finals as he had demanded, despite breezing through qualification to the tournament in Poland and Ukraine.
And now many are questioning his fate ahead of a crunch meeting with the France Football Federation on Thursday.
Blanc had been caught in the centre of a media storm amid reports of rifts between squad members, and has been scrutinised most for not taking harsh disciplinary action against Manchester City midfielder Samir Nasri, who was involved in altercations with French media on several occasions.
|BLANC HART LANE?
|5/1||Laurent Blanc is 5/1 with William Hill to be the next permanent Tottenham manager.|
"I have no information with regard to whether Blanc will stay as France coach," confided Dugarry to RMC Sport.
"If I know him well enough, I'd say Blanc will only want to stay on his own terms. He will want to know that he has the full support of the federation even in difficult times, with the power to do things as he wishes.
"If extending [his contract] does not give him such power, it's unlikely he'll stay."
Tottenham chairman Daniel Levy is a keen admirer of the French coach, and is rumoured to be considering him as a candidate to succeed Harry Redknapp.