Schumacher has been in a coma since the accident on December 29th and a family statement released on Thursday said he was "still in a waking up process" which could take a long time.
The statement came a day after Germany's biggest-selling newspaper, Bild,reported that Schumacher had contracted a lung infection, and days after rumours that he had died swept social media.
The statement said: "Michael's family would like to again express their sincere thanks for the continuous sympathy coming from all over the world. The good wishes they receive help the family and, we are convinced they also help Michael, who still is in a waking up process.
"As often in such situation, no day is like the next. The family is thankful for one's understanding that they would not wish to disclose medical details in order to protect Michael's privacy. As assured from the beginning we will continue to communicate any decisive new information on Michael's health state. We are aware that the wake up phase can take a long time.
"The family continues to strongly believe in Michael's recovery and place all their trust in the doctors, nurses and nursing auxiliaries team. The important thing is not the speed of the recovery but that Michael's healing process progresses in a continuous and controlled way."
Schumacher's spokesman announced on January 30 that the drugs used to keep him in a coma were being reduced with a view to bringing him back to consciousness. It was not announced when that process had begun.
Schumacher has been in intensive care in Grenoble University Hospital since being flown there by helicopter following his December 29th accident in the Alpine resort of Meribel.
The German fell and hit his head against a rock after venturing on to a small off-piste section while skiing with his son and a group of friends.
The impact of the collision was sufficiently powerful to split the helmet he was wearing.
Surgeons said he suffered bleeding and bruising on his brain and a scan showed "widespread lesions". After the surgery he was placed in a medically induced coma and his body temperature was cooled to reduce the risk of further damage.
Further surgery was carried out on December 30th to reduce bleeding.