The crash occurred at about 3 a.m. in Mwingi, Capitol FM News in Nairobi reported. The bus was traveling from Nairobi to Garissa to the northeast and was about halfway to its destination.
Traffic Commandant Samuel Kimaru said the driver lost control of the vehicle, which left the road, rolling over. Eleven people were killed immediately.
Getting those trapped under the bus out took about 4 hours, Kimaru said.
"It was a very bad accident," he said. "It took a lot of time for them to be rescued because the bus was badly damaged."
Twenty people were being treated at Mwingi District Hospital, Kimaru said.
During the weekend, 17 people were killed in two crashes. Twelve died when a bus bound for Mombasa from Nairobi collided head on with a tractor-trailer and another five died in a crash involving a van.