The bus was going to Bhimduttanagar from Melauli of Baitadi in western Nepal.
The accident happened at around 5 am on the hilly roads at Dhapeni of Chure Rural Municipality on Dadeldhura-Bhimdutta Highway.
According to the police, those killed include an elderly woman and a teenager.
Two of the seriously injured passengers have been to taken to Haldwani in the neighbouring Indian state of Uttarakhand for treatment, while the remaining 22 are undergoing treatment at the Padma hospital in Attariya, they said.