The boat was reportedly carrying about 70 illegal migrants, mostly Rohingya Muslims, to Malaysia when it sank in the Bay of Bengal early yesterday.
Border commander Zahid Hossain said no bodies had been recovered, but he quoted several survivors as saying they saw some bodies after the boat sank off Bangladesh's Teknaf coast, 200 miles south of the capital Dhaka.
Another coastguard official said survivors had said they were travelling to look for jobs in Malaysia.
The officials said other fishing boats rescued many of the survivors.
Dangerous attempts to go abroad for work – often through unscrupulous human traffickers – have lured poor youths in recent years, with local reports saying stateless Rohingya people living in Bangladesh often attempt the risky trips.
More than 25,000 Rohingya people live in two official camps run jointly by the government and the United Nations in Cox's Bazar. But hundreds of thousands of others live outside the camps.