At least 39 more 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. The reason for the panic was not immediately clear.
“As kids were going home from school there was a stampede as they were going down the stairs,” a spokesman for Red Cross Kenya Peter Abwao said, adding that “it’s a three-story building, it’s a classroom block.”
Anxious parents waited outside the Kakamega hospital for news of their children, Daily Nation pictures showed.
"My heartfelt condolences to parents that have lost loved ones in the unfortunate and regrettable tragedy at Kakamega Primary School. I wish a quick recovery to the injured children and pray that God grant strength to the affected families," former Prime Minister Raila Odinga said on Twitter.