The bus, carrying at least 45 passengers, veered off a narrow road in Jajarkot region in the morning, police officer Sher Bahadur Chaudhary said.
"At least two dozen passengers are feared dead," Chaudhary told AFP from Jajarkot.
With the bus completely under water, divers were searching for passengers in the Bheri river as police officers scoured its banks, Jajarkot police chief Dinesh Raj Mainali said.
"Five bodies have been found so far, including those of two children," Mainali told AFP from the scene.
Police have taken ten injured passengers to a nearby hospital for treatment, he said.
Police official Chaudhary said he feared the number of people on board may have been higher than the official passenger list since drivers often pick up extra travellers along the route without registering them.
Accidents are common on Nepal's highways because of poor roads, badly maintained vehicles and reckless driving.
At least ten people, including a Russian woman, died this month when two buses collided along a national highway in central Nepal.
Last month a crowded bus plunged down a hill in a town outside Kathmandu, killing at least 14 people, including two Israelis.
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