The plane, operated by US-Bangla, a Bangladeshi airline, veered off the runway while landing on Monday afternoon, causing a blaze that firefighters have been tackling.
About 67 passengers were thought to be on board, officials told the BBC. The number of casualties is unconfirmed.
Seventeen people on board have been rescued so far, the authorities say.
Photos and video posted on social media showed smoke rising from an airport runway.
The plane was identified in local media as S2-AGU, a Bombardier Dash 8 Q400, but this has not been officially confirmed.
The flight landed at TIA airport, also known as Kathmandu International Airport, at 14:20 local time (08:35 GMT), according to flight tracking website FlightRadar24.