The police said rescuers pulled bodies from the charred wreckage of the plane, operated by Bangladeshi airline US-Bangla, after a raging fire was put out.
The airline has blamed air traffic control, but the airport says the plane approached from the wrong direction.
Flight BS211 veered off the runway while landing on Monday afternoon.
The exact cause of the crash remains unclear and Nepalese Prime Minister KP Sharma Oli promised an immediate investigation.
However a recording of the conversation between the pilot and air traffic control minutes before the plane crashed suggests some misunderstanding over which end of the sole runway the plane was cleared to land on.
Moments before the plane crash-landed, an air traffic controller is heard in the recording obtained by the BBC from air traffic monitor LiveATC telling the pilot: “I say again, turn!”
Twenty-two people are being treated in hospital for injuries, police spokesperson Manoj Neupane told the BBC’s Nepali service.
The plane, which was flying from the Bangladeshi capital Dhaka, was a Bombardier Dash 8 Q400 turboprop and was 17 years old.
The plane landed at 14:20 local time (08:35 GMT), according to flight tracking website FlightRadar24.
A survivor said the plane had begun to behave strangely as it approached Kathmandu
“The aircraft was permitted to land from the southern side of the runway flying over Koteshwor, but it landed from the northern side,” Sanjiv Gautam, director general of the Civil Aviation Authority of Nepal, was quoted as saying by the Kathmandu Post.
“We are yet to ascertain the reason behind the unusual landing.”
However, US-Bangla Airlines chief executive Imran Asif blamed Kathmandu air traffic control.
“There were wrong directions from the tower. Our pilot was not at fault,” he told reporters at his office in Dhaka.
“Our pilot is an instructor of this Bombardier aircraft. His flight hours are over 5,000. There was a fumble from the control tower.”
Airport general manager Raj Kumar Chettri told Reuters news agency that the plane hit the airport fence before touching ground.
“The plane should have come from the right direction,” he said.
One of the survivors, Nepalese travel agent Basanta Bohora, described from his hospital bed what he had experienced.