The inquiry stems from an incident during Toulouse's Ligue 1 fixture against Le Havre on November 2, where Donnum, who has denied his actions were racist, waved his hand in front of his nose in the direction of Le Havre's Cameroonian midfielder Simon Ebonog, who is Black.
"It is important to make it clear that the justice system will pursue all acts of racism," Toulouse prosecutor David Charmatz told Reuters.
"The Justice Minister recently issued a circular stressing the need to combat such offences, which undermine the shared values of our Republic."
Le Havre manager Didier Digard strongly criticised Donnum's actions following the match.
"What other interpretation is there? If someone tells me it wasn't racist, then what is it?" he said.
The prosecutor stated that the investigation was launched independently "following the disclosure of the facts by Mr Digard."
Donnum said after the match that his reaction "had nothing to do with racism" and Toulouse FC released a statement to condemn "with the utmost firmness the unfounded accusations."
The Professional Football League's (LFP) disciplinary commission is set to rule on Donnum's conduct on Wednesday.