Schumacher was moved on Monday after emerging from a coma following his devastating ski accident in December.
The paper said he had his eyes open for most of the journey by private ambulance, which was conducted like a secret service operation.
Michael Schumacher did not speak, but he communicated by nodding his head during the 200-kilometre (125-mile) journey, the paper said.
The ambulance company was not told the identity of its world-famous passenger in advance, and staff had to give up their mobile phones when they came into contact with him.
The paper said Michael Schumacher had lost a lot of weight during his 170 days in hospital in the French city of Grenoble.
It tracked down the Swiss ambulance company, whose head Mathias Volken confirmed its involvement, but refused to give further details.
The seven-time world champion's spokeswoman Sabine Kehm on Monday thanked the doctors in Grenoble" and asked for "understanding that his further rehabilitation will take place away from the public eye".
The hospital where he is now being treated is one of Switzerland's best and has renowned neurology experts.
It is in the city of Lausanne, near where Michael Schumacher's wife Corinna and two children live.