The boat carrying some 60 passengers capsized after the strong current overturned it off the southeastern Kutubdia island.
"A total of 39 passengers have so far been rescued and two of our vessels are scouring the sea in search of more," a coastguard official told TV channels.
Some people managed to swim to safety.
A private channel quoting a navy official said divers traced the boat near the shoreline and found no body inside.
Police said it appeared all the passengers were Bangladeshi who intended to go to Malaysia illegally defying previous warnings against such attempts.
Bangladesh in recent years witnessed attempts by its nationals as well as Myanmar's Rohingya refugees to go to Malaysia for a better life.
Some 100 people have died in an identical incident in 2012.
Officials say middlemen take money from these people, promising them send them to Malaysia, but in most cases they land in jails there as intruders.
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