The bus was coming from Garhwal in the north and was headed to Ramnagar, with at least 42 passengers on board, Devendra Pincha, a local police officer told Reuters by phone.
The death toll is likely to rise as the condition of a few of the survivors is critical, he said.
Visuals from ANI news agency, in which Reuters has a minority stake, showed an overturned bus lying beside a river at the base of a hill.
State Chief Minister Pushkar Singh Dhami said rescue officials were working rapidly to evacuate the injured.
RHM/