The Arsenal official is Arsene Wenger's eyes in France and across Europe, and he's been keeping a watchful eye on the talented Frenchman currently playing his football for Real Sociedad.
Chelsea, Paris Saint Germain and Manchester United have all been linked with Griezmann as well, but it doesn't look like Arsenal will be making a bid anytime soon if Grimandi's comments are anything to go by.
He said: "He clearly has something. There is no question about that. But in the Champions League during very high level matches, he suffered. Notably against Manchester United."
In 40 appearances this season Griezmann has 20 goals and three assists for Real Sociedad, and he has had scouts from Europe's top clubs flocking to the Basque Country since his stunning goal against Lyon in the Champions League play-off round.
Chelsea, Arsenal and United were among a number of Premier League clubs reportedly keeping tabs on him, with United scout David Friio saying he believed Griezmann could be a key player for France at this summer's World Cup.
The £20m-rated 22-year-old received his first cap for his country against Holland in a friendly earlier this week.