Some 43 passengers on the bus have been rescued, while efforts have been started to find the five who are still missing, the Indo-Asian News Service reported.
"The bus fell into the river Bhairabanki at Fulkushma. Most of the people swam to safety or were rescued. Five to six bus commuters are missing," Bankura Superintendent of Police Mukesh Kumar told IANS.
District Magistrate G.A. Ansari said the rescue effort has been hindered by rain.
"Divers have been deployed. There is too much of rain and rescue work is difficult," Ansari said.
The cause of the incident was not reported.