Forty migrants were rescued but search and rescue efforts are under way to find 13-15 people still missing in the incident off the Kefken district, the statement said. The area lies about 130 kilometres east of Istanbul.
One pregnant woman who was rescued was taken to hospital but lost her baby, state-run Anadolu news agency said.
Mehmet Unal, a local official in the nearby town of Kandira, said the boat was believed to be carrying 70 migrants, most of whom were Iraqis.
The boat reportedly sank due to bad weather and rough waters. Western Turkey and the Black Sea had been hit by a heavy storm and rains overnight.
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Turkey has emerged as a hub for migrants from trouble zones around the world including Syria, Iraq and Afghanistan who are seeking a route west to the European Union.
In 2015, around a million people undertook risky sea journeys to cross the Mediterranean to Europe, mostly to Greece from Turkey over the Aegean Sea.
The numbers of migrants crossing the Aegean has now dwindled. But increasing numbers, now rising to hundreds within a month, are now trying to cross the Black Sea from Turkey to EU member Romania.
Some 834 migrants were caught and 10 smugglers detained in seven Black Sea incidents between mid-August and early September, according to statistics published by Anadolu.
Separately, 237 migrants and five organisers were caught in two other incidents in the area whose timing was not specified.
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