District superintendent of police Mukesh Kumar said the bodies belonged to a 14-year-old boy and a 27-year-old youth.
The bus was on way to Durgapur from Jhargram in West Midnapore district and had about 65 passengers on board, including six people on its roof, police said.
The driver lost control over the vehicle when water entered its silencer and was washed away near Phulkulsma, he said adding most of the people inside the bus were rescued by the local people.
The driver and the conductor of the bus were among those rescued. The injured have been rushed to the local public health centre.
Disaster management group personnel, including divers from Durgapur have reached the spot, which is between Khatra and Raipur. Fire brigade, health team, local administration and police too were assisting operations.
Chief Minister Mamata Banerjee told reporters at the state secretariat in Kolkata that three or four passengers of the bus were missing and there was no passenger inside the bus.
Child development minister Shyamapada Mukherjee and the SP have been sent to the spot, she said.