The ship has arrived at the port of Cesme, near Izmir.
The embassy is unable to say when the Finns boarded the ship. There are 479 people on the ship, 321 of which were injured in the fighting in Libya. According to the Turkish Red Crescent, 70 of them are gravely wounded.
Apart from the wounded, 108 Libyans and 29 other national were fleeing Libya on the ship.
Embassy officials believe the Finnish-Libyan family, including children, boarded the ship in Benghazi.
Turkey is prepared to provide the travellers with medical assistance.
