The crash, shortly before midnight Monday, killed the driver and left several people with what emergency workers called "life-changing injuries," The Sun reported.
The bus crashed on the A3 highway in Surrey as it traveled to Merseyside from the Bestival music festival.
Police described the crash as a "major incident." No other vehicles were believed to be involved, the report said.
Inspector Richard Mallett described the scene as a "body recovery operation" and said many of the victims were still trapped inside the bus.
Surrey police believe the vehicle may have been overloaded, The Daily Telegraph reported.
The bus had seats for 52 people, including the driver, but 53 people have been accounted for, they said.
Mike Dower, manager of the bus company, MerseyPride, downplayed driver error as responsible for the accident. He said the driver, Colin Daulby, believed to be in his early 60s, had "a good rest" at his sister's house before returning to the road.
Daulby had been with the bus company only a few months, Dower said.