The twin propeller plane, a Bombardier Dash 8, can carry about 70 passengers, though it was not immediately clear how many people were on board.
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At least 38 bodies have been recovered with 10 people still unaccounted for, according to an AP journalist who arrived at the scene soon after the crash and saw the US-Bangla Airlines plane broken into several large pieces, with dozens of firefighters and rescue workers clustered around the wreckage.
An airline official said some injured passengers had been taken away by ambulance.
"I can't say exactly how many have been rescued," Mohammed Selim, the airline's manager in Kathmandu, told Dhaka-based Somoy TV station by phone.
Amanda Summers, an American who works in Nepal, watched the crash happen from the terrace of her home office, not far from the airport.
"It was flying so low I thought it was going to run into the mountains," she said. She said it was unclear if it had reached the runway when it landed. "All of a sudden there was a blast and then another blast," she said.
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Fire crews put out the flames very quickly, perhaps within a minute, she said, though clouds of thick, dark smoke rose into the sky above Kathmandu, the capital of the Himalayan nation.
Selim, the US-Bangla manager, said the plane had circled the airport twice as it waited for clearance to land.
US-Bangla Airlines operates Boeing 737-800 and smaller Bombardier Dash 8 Q-400 planes.
The airline, part of US-Bangla Group, is based in the Bangladeshi capital, Dhaka, and flies to several domestic and international destinations. The parent company is involved in a number of industries, including real estate, education and agriculture.
Kathmandu's airport has been the site of several deadly crashes. In September 2012, a Sita Air turboprop plane carrying trekkers to Mount Everest hit a bird and crashed shortly after takeoff, killing all 19 on board.