They say the boat sank after hitting a cargo vessel in the Surma River at Alipur, some 240km (150 miles) north-east of the capital, Dhaka.
More than 100 people – mostly labourers – are said to have been on board. Most of the victims were reportedly women and children.
Rescuers have been sent to the remote area to look for survivors.
“Villagers have recovered the bodies of 37 people and almost all of them are women and children,” Abul Moshem, a senior government official in Sunamganj district told the BBC.
“At least 20 people are still missing. We have rushed our rescue teams to the accident site.”
The boat sank after colliding with a much heavier trawler in darkness.
Officials said the accident might have happened due to poor visibility.
“Many male passengers who were sitting on the roof of the boat managed to swim to safety. While those passengers, mostly women and children, who were inside the cabin drowned in the river,” Golam Kidria, superintendent of police in Sunamganj district told the BBC.
Villagers using torch lights were carrying out immediate rescue operations.
Boat accidents are common in the south Asian country due to overloading and lax safety standards. – BBC News