The Yeti Airlines flight from Kathmandu to the tourist town of Pokhara crashed on landing, catching fire.
Video posted on social media shows an aircraft flying low over a populated area before spinning sharply.
There were 68 passengers on board, including at least 15 foreign nationals, and four crew members.
Some 200 Nepalese soldiers are involved in the rescue at the crash site in the gorge of the Seti River, just one and a half kilometres from the airport.
Video of the crash site show thick billowing black smoke and burning debris.
“We expect to recover more bodies,” an army spokesman told Reuters, saying the plane “has broken into pieces”.
Prime Minister Pushpa Kamal Dahal called an emergency meeting of his cabinet and urged state agencies to work on rescue operations.
Of the passengers, 53 are said to be Nepalese. There were five Indian, four Russians and two Koreans on the plane. There was also one passenger each from Ireland, Australia, Argentina and France among others.
In May 2022, all 22 people on board a plane operated by Nepali carrier Tara Air — 16 Nepalis, four Indians, and two Germans — died when it crashed.
In March 2018, a US-Bangla Airlines plane crash-landed near Kathmandu’s international airport, killing 51 people.
That accident was Nepal’s deadliest since 1992, when all 167 people aboard a Pakistan International Airlines plane died when it crashed on approach to Kathmandu.
Just two months earlier, a Thai Airways aircraft had crashed near the same airport, killing 113 people.