Seven times Formula One world champion Michael Schumacher is no longer in a coma and has left the Grenoble hospital where he has been treated since a skiing accident in the Alps five months ago, his spokeswoman said on Monday.

"Michael has left the CHU Grenoble [hospital] to continue his long phase of rehabilitation. He is not in a coma anymore," his manager, Sabine Kehm, said in a statement.

A spokesman for the University Hospital of Lausanne later confirmed Schumacher, who lives with his family in a town in western Switzerland between Lausanne and Geneva, had arrived on Monday morning but declined to say which unit he was being treated in.

Grenoble University hospital. Photograph: Abaca/Barcroft

The announcement came after a German magazine, Bunte, reported last Friday that the 45-year-old champion had been moved from the Grenoble hospital's intensive care department, where he had been in a medically induced coma, into a rehabilitation unit. The magazine said Schumacher was "out of danger", although his chances of a full recovery had diminished. It said preparations were under way to move him to a specialised rehabilitation clinic.

In Germany, the news was greeted euphorically. "What a great news!!! Get well soon Schumi!!! I'm so glad and happy when I just heard it!", tweeted Arsenal footballer Lukas Podolski, a few hours before the German team's first World Cup match.

On Saturday, the Germany forward had sent an emotional message to Schumacher from his team's training camp in Brazil, saying he hoped he could watch some of the games and that "if we get the title, we can bring him some happiness with it".

Schumacher at the Mugello racetrack in Scarperia, central Italy, in 2006. Photograph: Tony Gentile/Reuters

The Austrian former Formula One champion Niki Lauda told RTL television he was "incredibly happy" for Schumacher: "I hope for him that he managed to get through rehab as quickly as possible and join us again soon at Formula One."

Schumacher sustained life-threatening head injuries after falling on to a rock while skiing with friends in the resort of Méribel, in the French Alps, on 29 December. He underwent two operations in Grenoble to remove blood clots on the brain.

Monday's update was the first issued by Kehm since 4 April, when she said Schumacher was making small signs of progress and was showing "some moments of communication".

But since then, former Formula One doctor Gary Hartstein wrote on his blog earlier this month that he feared the worst, given the lack of information from Schumacher's family. He said he was "quite afraid (and virtually certain) we will never have any good news about Michael".

Kehm thanked the doctors and nurses at Grenoble hospital and the rescue services who took him from the scene of the accident. She said they had done "an excellent job in those first months".

She also expressed gratitude for messages of support and asked for privacy while Schumacher's rehabilitation took place.

"The family also wishes to thank all the people who have sent Michael all the many good wishes … We are sure it helped him," the statement said. "For the future we ask for understanding that his further rehabilitation will take place away from the public eye."

Schumacher has a home on the shores of Lake Geneva. It was reported in March that his wife Corinna had spent £10m on building a medical suite there.