They say about 35 people were rescued after the accident in the Meghna River, south-west of the capital Dhaka. Several bodies have been recovered.
Unconfirmed reports say the Shariatpur-1 hit by a small oil tanker.
Ferry accidents are common on Bangladesh’s vast river network and scores are killed every year.
“The death toll is likely to rise as more bodies are feared trapped inside,” local police chief Mohammad Shahabuddin Khan told the AFP news agency. “We will get a better picture of the casualties once the sunken ferry is pulled out of the water.”
The ferry was reportedly travelling to Dhaka from the Shariatpur district.
Most ferry accidents in the country are blamed on poor safety standards and overcrowding.
Last April, at least 23 people died after a ferry carrying more than 100 passengers capsized in the east of the country.
In June 2010, about a dozen people were killed after a packed ferry capsized in storms in north-east Bangladesh.
In November 2009, 118 people died in two ferry accidents within a week.