The boat caught fire on the Sugandha river near Jhakakathi, a rural town around 250km south of national capital Dhaka.
Those injured have been rushed to local hospitals, reported the Dhaka Tribune newspaper.
A video circulating on social media showed a three-storey ferry, called the MV Abhijan-10, engulfed in flames. The authenticity of the video could not immediately be ascertained.
Local police chief Moinul Islam told news agency AFP that casualties from the tragedy could rise, adding that some people died after they jumped off the boat and drowned.
“We have recovered 32 bodies. The death count may rise. Most died from the fire and a few by drowning after they jumped into the river,” he said.
The fire reportedly started at around 3am in the ferry’s engine room. Some 15 firefighting units were sent to the scene at around 3.50am and the blaze was brought under control at 5.20am, reported Bangladesh’s The Daily Star newspaper.
One of the survivors, Saidur Rahman, told the newspaper that the flames had spread rapidly when the vessel was near a bridge. He said there were 500 people on board, including children and the elderly.
“The engine room of the launch suddenly caught fire around 3am and spread rapidly when the vessel was near the Gabkhan Bridge,” he said.
“I caught a burning smell and came out of VIP cabin to find out that there was a fire. Me, my wife and brother-in-law then jumped in the cold water and swam to the bank,” he added.
A number of accidents involving ferries and boats in the country occur routinely, primarily because of overcrowding and poor safety measures.
In August, at least 22 people died after a passenger boat collided with two sand-laden bulkheads in the Brahmanbaria district, around 82km east of Dhaka.
Earlier in April, 34 people died when a double-decker ferry sank after it was hit by a cargo vessel in the Shitalakkhya River in Narayanganj district near Dhaka.