"There are 11 bodies," Alexei Kravchenko, spokesman for Russia's Federal Agency for Maritime and River Transport, told AFP news agency.
"Another three people went under water before rescue workers' eyes," he said, adding that they are most likely dead.
The ships, the Kandy and the Maestro, had crews of nationals from India and Turkey. The Kandy had a 17-strong crew, while the Maestro carried 14 people.
Twelve people survived, Kravchenko said, but "no one knows where the other five people are."
The fire broke out when one vessel was transferring fuel to the other, driving both crews to jump overboard, said Kravchenko. "A search-and-rescue operation is continuing," Kravchenko added.
An industry source told Reuters news agency there were stormy conditions in the sea when the incident happened.
The accident struck close to the Kerch Strait, which has become a new flashpoint in tensions between Russia and Ukraine.
The strait between Russia-annexed Crimea and southern Russia controls access from the Black Sea to the Azov Sea, where there are both Russian and Ukrainian ports.
In November, Russia arrested three Ukrainian navy vessels and their crews in the Kerch Strait, raising tensions between the two countries.
The confrontation was the first open military incident between Kiev and Moscow since 2014, when Russia annexed Crimea.