A passenger bus was transporting people from Botswana, a landlocked country in Southern Africa, to Moria, a town in Limpopo
At least 39 more pupils were badly hurt in the incident at the Kakamega Primary School, the Daily Nation newspaper said, citing Western Region Police Commander Peris Kimani.
"We lost 14 of them," Education Minister George Magoha told Citizen TV. "One life (lost) is a life too many."
The Daily Nation said that some of the children fell from the third floor as they ran.
"As kids were going home from school there was a stampede as they were going down the stairs," said Peter Abwao, a spokesman for Red Cross Kenya. "It's a three-storey building, it's a classroom block."
Anxious parents waited outside the Kakamega hospital for news of their children, Daily Nation pictures showed.