The airport was closed on Friday after the incident. All arriving flights were transferred to other nearby countries while outgoing flights were cancelled.
According to officials two people were injured.
"Our teams are working to remove the plane and reopen the airport," the airport's general manager Raj Kumar Chettri, said as quoted by AFP.
There has been continuous rainfall in the airport's area since 11 July, which could have been a factor in the incident.
In September last year, it took more than 10 hours for Nepalese authorities to remove an aircraft from the runway after a similar incident.