The Comoros-registered Raptor had a crew of 14. Only one survivor has been rescued so far.
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The search continues for 12 crew members missing after a cargo ship sank in rough weather on Sunday.
Frontex, the European Union's border and coastguard agency, has joined rescue efforts near the Greek island of Lesbos.
The Comoros-flagged Raptor was transporting salt from Alexandria, Egypt, to Istanbul and issued a distress signal before sinking.
It had a crew of 14, including eight Egyptians, four Indians and two Syrians, the Greek coast guard said.
A 40-year-old Egyptian crew member was rescued on Sunday while the body of another Egyptian member of the crew was recovered by a commercial vessel assisting in the search.
The Raptor first reported mechanical problems at 07:00 Sunday morning, and sent a distress signal at 08:20.
Shortly after that it disappeared around 8 kilometres southwest of Lesbos, according to the authorities.
Frontex, which patrols the Lesbos area monitoring for illegal migration, sent a surveillance aircraft and a patrol boat to join the Greek coast guard search.