Russia confirmed earlier that the hull of the Liman, part of its Black Sea Fleet, had sustained a breach, with crew working to keep it afloat.
It is not yet clear what caused the collision.
The ship hit a Togo-flagged boat carrying livestock, Turkish media say.
Turkish Prime Minister Binali Yildirim called his Russian counterpart, Dmitry Medvedev, to express his sadness over the collision, sources in the Turkish prime minister's office were quoted by Reuters news agency as saying.
The Russian Black Sea Fleet passes through the Bosphorus Strait for deployments in the Mediterranean, notably in Syria.