Three passengers were rescued as the continuous rain made rescue efforts difficult.
The survivors were being treated in the hospital, government administrator Khima Nanada Bhusal said, adding they reportedly jumped out of the bus and swam to the banks, where locals found them and took them to a nearby hospital.
Landslides also blocked routes to the area in several places, according to Bhusal.
Additional rescuers and security forces have been sent to help with rescue efforts.
The buses were swept off the highway about 3am on Friday near Simaltal, about 100km west of the capital, Kathmandu.
One bus was carrying at least 24 people, while the other had at least 42, but more could have boarded, Bhusal said.
A third bus was hit by another landslide on Friday morning a short distance away on the same highway, killing the driver, Bhusal said.
It was not clear if there were any other casualties.
Prime Minister Pushpa Kamal Dahal said he was saddened by the news and expressed concern over recent flooding and landslides.
Several government agencies were searching for the missing, he said in a post on the social media platform X.
On Thursday night, a landslide buried a hut and killed a family of seven near the resort town of Pokhara.
The family were asleep when the landslide crushed their hut and damaged three more houses nearby.
Monsoon season brings heavy rains to Nepal from June to September, often triggering landslides in the mountainous Himalayan country.
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