Rennes holding midfielder M'Vila injured himself in the first minute of the 2-0 friendly win over Serbia last Thursday. He was feared to be a doubt for the entire tournament but will remain with the France squad.
A statement from the French Football Federation said: "The MRI confirmed that it's a moderately serious sprain, without aggravating factors.
"It's running its course, there are no complications."
Paris Saint-Germain midfielder Matuidi will definitely miss the encounter with the Three Lions on June 11 after suffering a minor tear to his right hamstring.
Goalkeeper Steve Mandanda is fit however for the England game, after recovering from a toe injury.
France face their final Euro 2012 warm up match when they meet Estonia in Le Mans on Tuesday.