L'Equipe said the awakening process, which could take several days and even weeks, actually began on Tuesday, led by the Grenoble doctor Emmanuel Gay.
The report said Schumacher had initially "responded positively" to the process.
Actually, L'Equipe was not the first to report the story, after the respected Le Journal de Dimanche said on Sunday that the beginnings of the awkening process were set to begin.
And, crucially, the stories were reportedly confirmed by the Grenoble neurosurgeon Stephan Chabardes, according to French radio RMC, and BFMTV television.
Germany's Welt newspaper asked Kehm about Chabardes' reported confirmation, but Schumacher's manager referred only to her new media statement about "speculation".
And Bild newspaper reported that a new memo was sent to all employees at the treating Grenoble hospital, warning them not to disclose information "about the patient M Schumacher".
L'Equipe deputy chief editor Jeremie Arbona, however, told Bild: "We cannot disclose our sources, of course, but they are reliable."
No matter what the truth is, Schumacher's situation remains severe. A leading German neurosurgeon, Andreas Ferbert, is quoted by the newsmagazine Focus: "Dcots put a patient in a coma for three or four weeks only when the condition of the brain is very serious.
"Four weeks is a very long time," he explained.