"Twenty-nine passengers have been killed in the crash. Twelve passengers who have been seriously injured are being treated in hospital, 50 kilometres (31 miles) from the site," said Air.
The bus had 35 passengers registered at its last checkpoint but drivers on rural routes tend to pick up passengers on the way and police suspect the death toll could rise much higher when the crash site in Chhatiwan village is searched.
"The area where the accident occurred has light forest cover... We don't know the cause of the accident yet but locals have told us that it was foggy during the night," Air said.
Police were called when a man driving by the crash sight heard the screams of survivors, around half an hour after the crash, he added.
"We have deployed 50 policemen but they have told us the rescue efforts have been hampered because the bus is stuck on a precipice."
Accidents are relatively common on Nepal's highways because of poor roads, badly maintained vehicles and reckless driving, and death tolls running into double figures are frequent.
In September last year 27 people were killed when a bus veered off a mountain road in western Nepal, while 38 pilgrims died on an overcrowded bus carrying them from India to a Hindu festival in southern Nepal two months earlier.