Five more sailors are misssing.
Russian media reported that one of the vessels was hit by an "explosion" and people on board jumped into the water to escape the blaze.
The Federal Agency for the Sea and River Transport said the fire erupted while fuel was being pumped from one tanker to another.
The blaze then spread from one ship to the other, according to Russian news agencies.
Rescuers pulled 12 people alive from the water and recovered 11 bodies.
A further three bodies were spotted but were not recovered.
The rescue mission is ongoing.
Both ships, the Maestro and the Candy, sail under the Tanzanian flag.
In November, Russia detained three Ukrainian navy vessels and their crews in the Black Sea near the Kerch Strait, fuelling tensions between the two countries.
Relations between the nations have remained volatile since Russia annexed Crimea in 2014.
Additional reporting by agencies
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