Three survivors apparently swam to safety, but rescuers by late morning on Friday had not found any trace of the buses, which likely were submerged and swept downstream in the Trishuli River.
Nepal's rivers generally are fast-flowing due to the mountainous terrain.
Heavy monsoon downpours in the past few days have swollen the waterways and turned their waters murky brown, making it even more difficult to see the wreckage.
The buses were on the key highway connecting the capital to southern parts of Nepal when they were swept away about 3am near Simaltal, about 100km west of Kathmandu.
More landslides blocked routes to the area in several places, government administrator Khima Nanada Bhusal said.
Additional rescuers and security forces were sent to help with the rescue efforts.
Police and army personnel were searching using rubber rafts.
Divers with scuba gear were also dispatched, according the Chitwan district police.
One bus was carrying at least 24 people, while the other had at least 42, but more could have boarded en route, Bhusal said.
The three survivors were being treated in the hospital, 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.
A third bus was hit by another landslide on Friday morning a short distance away on the same highway.
Bhusal said the driver was killed but it was not clear if there were any other casualties.
Nepalese 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.
