The bus was traveling on the Kullu-Ani road Wednesday when it skidded, falling into the turbulent Beas River at Jhiri district in the mountainous Himachal Pradesh state in northern India, authorities said.
The Indian Express newspaper reported rescue teams had retrieved 32 bodies and rescued 24 people trapped in the wreckage of the bus. Several more passengers were yet to be accounted for, the report said.
Most of the victims were women and school students returning home.
The Times of India reported there were about 65 passengers in the bus, with several of them riding on the vehicle's roof, a common sight on Indian roads. It said at least 32 people died, 25 more were injured and the rest were missing.
Rescuers included local river-rafting operators who managed to reach the partially submerged bus and retrieve the bodies and rescue the trapped passengers, the report said.
A police official said some of the missing may have drowned. Rescue operations continued Thursday morning.
Road accidents in India are common, claiming tens of thousands of lives every year.