The authorities received a distress signal early in the morning from a fishing vessel floating near Pharmakonisi island, in the southeast Aegean.
The vessel is believed to have run into a mechanical problem in rough seas.
As the Greek coastguard boats approached, two migrants jumped into the water for unknown reasons. The other passengers, in their effort to save them, made the vessel capsize.
According to the rescued migrants, 12 others are still missing. Greek authorities are still searching the area, near the Turkish coast, with the assistance of a Super Puma helicopter.
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In November, at least 12 migrants of Syrian origin, with four children among them, drowned in the Ionian Sea when their vessel capsized.
Greece is one of the main ports of entry into the European Union for migrants and refugees fleeing war-torn and impoverished countries in Africa, the Middle East and the Indian subcontinent.
Refugee traffic has soared over the past year because of the ongoing war in Syria. Arrivals by sea have increased owing to stricter controls on the Greek-Turkish northern land border.