"It is a horrible scene. Bodies are strewn all over," traffic police official Samuel Kimaru said by telephone from the scene near the town of Narok.[break]
"It is difficult to tell exactly what happened but all indications point to speeding and possibly overloading," he said. "It is quite disturbing."
The Kenyan Red Cross said it had confirmed 41 deaths and 27 others with "multiple injuries".
The bus was travelling from Nairobi to Homa Bay on the banks of Lake Victoria.
It was about halfway into the journey when it veered off the road and plunged into a valley, rolling over several times.
The Red Cross said the accident occurred shortly after 2:00 am (2300 GMT Wednesday). Pictures from the organisation showed the bus in the bushes at the bottom of a hill, its roof ripped off.
Kimaru said police had "a difficult time recovering the bodies because this place is hilly and bushy".
"We´ve dozens others injured and we are not certain how many because they were taken to various hospitals... we have officers checking with the hospitals," Kimaru said.
Related story
Nepal wins T20 Int'l Cricket Series against Kenya
