Grenoble prosecutor Jean-Yves Coquillat told AFP the investigation was launched on Friday following a complaint from the hospital where Schumacher was treated for months after the December crash.
"For now nothing is certain in this case," he said.
"The perpetrator or perpetrators of the theft contacted some French, Swiss and German journalists. An individual, communicating by email, is asking for 60,000 Swiss francs (49,000 euros, $67,000) to provide a document of several dozen pages that summarises everything that happened at Grenoble during Michael Schumacher's hospitalisation," Coquillat said.
He said parts of the document were provided by email to prove it was legitimate.
A source close to the investigation said the document appeared to be a summary written by Schumacher's doctor for his transfer to Switzerland this month after emerging from a coma.
Police were tracing who had access to the document and the hospital was checking its computer system to see if it may have been hacked, sources said.
The ex-Formula One star's management said Monday that the documents had been stolen and were being offered for sale.
His spokeswoman Sabine Kehm warned that the documents were confidential and promised to file suit against any publication that releases the medical records.
Schumacher had been treated since December 29 in the French Alpine city of Grenoble after he slammed his head against a rock while skiing with his son and friends.
The racing star underwent two operations to remove life-threatening blood clots after the freak accident that shocked the world, before being plunged into a medically induced coma.
The 45-year-old is undergoing further treatment at a hospital in the Swiss city of Lausanne.