The bus, packed with villagers, fell 200 metres (650 feet) into a river in Jajarkot, 250 km (220 miles) west of Kathmandu.
The injured were evacuated to a local hospital, police official Hari Bhakta said. The cause of the incident was not immediately known.
22 dead as bus plunges into Nepal river
The nearest hospital was also struggling to cope with the number of casualties and some people were being taken to other districts for treatment.
The officials said that there were around 50 passengers on the bus, blaming the accident on overcrowding.
However, the bus was also said to be in a poor state which was travelling down a narrow and windy road from the village of Khalanga in Jajarkot district to Khara in Rukun.
Road crashes, which are common in Nepal, are blamed on poor road conditions and ill-maintained vehicles.
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