There were no fatalities, China's Xinhua News Agency reported.
The plane, with a three-member crew and 18 passengers, had taken off from Pokhara, 200 kilometers (125 miles) west of the Himalayan country's capital Kathmandu, and was proceeding to Mustang near the border with Tibet.
At Jomsom airport in Mustang, the plane skidded off the runway and fell into the Kali-Gandaki River, Xinhua reported. The passengers included eight Japanese nationals and 10 Nepalese, the news agency said.
"One of the wheels of the plane had problem while landing. The plane skidded off the airport and fell into the Kali-Gandaki River," Xinhua quoted a witness as saying.
The extent of the injures to those aboard was not immediately known, but Nepal police spokesman Keshav Adhikari told Xinhua those with serious injuries were being flown to Pokhara. Others would be treated at a hospital in Mustang.
Nepal's Himalayan newspaper, quoting RSS news agency, said the seriously injured included all three crew members and three passengers.
Mustang is popular with foreign trekkers and Hindu pilgrims visiting the Mount Annapurna.
The cause of the incident was being investigated.