The incident took place 265 km from Lucknow.
While eight bodies have been fished out of the Hazara canal and 22 passengers have been saved from the canal waters, police said they fear more than two dozen people have been washed away.
"The bus was jam-packed and we are told that more than 58 passengers were travelling in it," a senior official overseeing the rescue operations told IANS.
District Magistrate Lokesh M told IANS that the condition of some of the passengers rescued from the Hazara canal continues to be critical.
"Rescue operations are on as we have not been able to locate the missing passengers," he said.
The injured passengers told police that the bus driver was trying to overtake a truck near Pilua police station when he lost control of the vehicle and it fell into the canal.
Home department officials said divers have been put into service to locate more passengers who might be trapped in the canal.
An official said it would only be later in the day that identification of those dead would be possible.