The Admiral Kuznetsov is visiting Turkey with a group of Northern Fleet warships, which have been holding naval exercises in the Mediterranean since the start of 2009.
"A fire occurred on board the aircraft carrier, and one crew member was killed. The circumstances of the incident are being established," the source said.
According to preliminary reports, the fire was caused by a short-circuit, and the sailor was killed by carbon monoxide poisoning. An investigation is underway.
The incident did not affect any technical systems on board the warship, the source said.
He added that up to 90% of shipboard fires around the globe were caused by electrical malfunctions.
The Russian Navy has so far made no official comment.