The boat sank 12 nautical miles southwest of the island, according to ERTNEWS, which reported the 40 missing, AFP reported.
A huge rescue operation is under way in the sea south of the island of Gavdos after the boat capsized shortly after midnight, the coast guard said.
An Italian frigate and helicopters were operating at the site, with more ships heading to the area.
Local media said another man was taken by helicopter to hospital in the city of Chania on Crete and was admitted to intensive care.
Greece has seen a 25-percent increase overall this year in the number of migrants entering, fleeing war and poverty, with a 30-percent increase to Rhodes and the southeast Aegean, according to the migration ministry.
Several similar accidents have struck in recent weeks. In the last one in late November, eight migrants, six of them minors, died north of the island of Samos, on a route frequently used by people smugglers.