Two coastguard boats and a helicopter were searching chilly winter seas off the Turkish holiday resort of Marmaris, but only one Iraqi survivor had so far been found, an official at the town's coastguard said.
"We have not found anybody yet but rescue efforts will continue through the night as long as weather permits," a coastguard official told Reuters.
The vessel was carrying Iraqi, Afghan and Jordanian migrants, another coastguard official said.
A passing ferry picked up the Iraqi survivor while search and rescue teams had only found the boat's tiller and an empty life jacket.
Marmaris is within sight of the Greek island of Rhodes in clear summer weather, but the waters in the area where the Aegean and Mediterranean Seas meet are notoriously treacherous.
Greek Merchant Marine Ministry officials in Athens said their search and rescue teams had not found anything.
"We have not been asked for help officially by Turkey. Greek coastguard at the island of Rhodes sent out vessels but no drowned bodies were found within Greek waters. Search and rescue operations have stopped," officers on duty at the ministry's central search and rescue operations told Reuters on Sunday.
They said the incident began unfolding late on Saturday.
"It must have taken place deep within Turkish waters, we're not engaged at all," said officer Lucas Maheras.
Each year thousands of migrants mainly from eastern Turkey, neighbouring Iraq and Iran try to enter Greece and other EU countries on small boats from Turkey's long coastline.
Many are caught, some get through and an unknown number drown.