Since the boat capsized near the Myanmar side of the border, Jalil said many may have swum to the Rakhine coast.
The exact number of people onboard the boat which capsized on Sunday remains unclear, with Bangladesh border guard officials giving estimates varying from 40 to up to 100.
"It capsized near (the coastal village of) Galachar with almost 100 people".
Last night, their vessel turned over and sunk off the southern shores of Bangladesh, the authorities said.
So far it is thought about 150 people have drowned whilst travelling in small boats not fit for rough journeys.
Border Guard Bangladesh (BGB) official Abdul Jalil told AFP 12 bodies had been recovered after an all-night rescue operation, saying "they include 10 children, an elderly woman and a man".
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The cause of the boat's sinking has been attributed to high waves and bad weather.
The refugee crisis erupted after Rohingya militant raids on Myanmar police posts on August 25 prompted a brutal military backlash.
Last month, almost 60 Rohingya were killed when another boat sank.
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The United Nations has said the army campaign could amount to ethnic cleansing while Myanmar military leaders have blamed the unrest on the Rohingya.
They are presumed to have crossed the Naf River on boats.
Bangladesh's Prime Minister Sheikh Hasina says her government will continue to support the Rohingya but Myanmar must take them back.
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