The accident took place at Nausehri, some 45 kilometres (27 miles) north of Muzaffarabad, the capital of Pakistan-controlled Kashmir, when the minibus driver lost control.
The vehicle veered off a road, falling 100 metres (110 yards) into the gushing river, local government and rescue officials said.
“More than 23 people were killed and three others wounded in the bus accident,” local government official Ashfaq Gilani told AFP.
“We have only found three bodies and three wounded people. More than 20 other passengers and the wreckage of the bus have been swept away by the river,” he added.
“We have presumed them dead as there is no chance of their survival,” Gilani said.
Rescuers said the terrain was difficult and that they would resume the search for the remainder of the passengers at dawn on Saturday.
Pakistan has one of the world’s worst records for fatal traffic accidents, blamed on poor roads, badly maintained vehicles and reckless driving.