The 2 a.m. accident occurred on the Nairobi-Narok highway in the Ntulele area, The Standard reported.
The driver crashed through the guardrail at a sharp curve, flipping over the bus and tearing off the roof, the Kenya Red Cross said in a statement.
"From our investigations, the driver did not even attempt to brake or negotiate the corner," Traffic Commandant Samuel Kimaru said. "He seemed to have slept."
The driver is believed to be alive and authorities were trying to locate him, the New York Times said.
Witnesses said rescue operations took some time as passengers were trapped under the overturned bus after it fell into a ditch.
Rescue operations were led by the Kenya Red Cross and police.
Rural roads in Kenya are often unlit and accidents are frequent, with drivers speeding on narrow, two-lane stretches, the times said.
"We are losing too many Kenyans through road accidents and the onus is us as car owners, as commuters, as traffic police and as leaders," President Uhuru Kenyatta wrote in a statement on Twitter on Thursday.