One of the ships, the Candy, had a 17-member crew, including nine Turkish citizens and eight Indian nationals.
The other one, the Maestro, had 15-member crew, including seven Turkish nationals, seven Indian citizens and an intern from Libya, Russian news agency Tass quoted maritime authority as saying.
At least 11 sailors have died, Russian Maritime Agency said was quoted by RT news, a Russian television network.
"Presumably, an explosion occurred (on one of the vessels). Then the fire spread to another vessel. A rescue tug is en route," said a spokesman for the Russian Maritime Agency.
Some three dozen sailors managed to escape the burning ships by jumping off the vessels.
Twelve persons have so far been rescued from the sea. Nine sailors are still listed as missing, the spokesperson said.
Severe weather conditions at sea have prevented rescue ships from taking victims to the shore for medical treatment, the report added.
The Kerch Strait is a key waterway that holds strategic importance for both Russia and Ukraine.
It is an important economic lifeline for Ukraine that allows ships leaving the port city of Mariupol to access the Black Sea.
It's also the the closest point of access for Russia to Crimea, a peninsula Moscow annexed in 2014. A Russian-built bridge over the Kerch Strait opened in May last year.