The boat is believed to have sunk Saturday off the coast of Sererihisar, south of the city of Izmir, Anatolia reported, quoting local governor Orhan Sefik Guldibi. Six of the migrants were rescued late Sunday.
Guldibi said rescuers had recovered 20 bodies that had washed ashore.
"The survivors said there were between 60 and 70 people on board. We are afraid that the death toll will rise," he told Anatolia.
There was no immediate information on the migrants' nationalities.
Thousands of immigrants from Asia and Africa enter Turkey illegally on their way to European countries in search of jobs and a better life.
Smugglers often take the migrants to Greece on fragile and overcrowded boats, and such accidents are not uncommon.
The European Union, which Turkey wants to join, is pressing Turkey to crack down on illegal immigration.
The choice of the city of Helsinki is not incidental as the capital of Finland had hosted US-Soviet negotiations on the limitation of nuclear stockpiles in 1969