The dead "may have been crew members of [a] dry cargo ship Nakhichevan that sank in the port of Kavkaz," a spokesman told the Interfax news agency.
Three of four freighters that also sunk during the same storm were carrying sulphur and the fourth contained scrap metal, officials said.
The hull of another tanker had cracked but the ship remains afloat.
Another six other ships have also run aground.
Russian officials said shipping in the area had been warned in advance of heavy storms but it appeared some captains had chosen to ignore the advice and put to sea.
The tanker Volganeft-139 was on its way from the port of Azov in the southern Russian region of Rostov to Kerch in Ukraine's eastern Crimea when high waves broke its hull early on Sunday, Russian media reported.
The tanker, designed primarily for rivers and in service since 1978, was carrying 4,000 tonnes of fuel oil as it traversed the Kerch Strait, a waterway running between the Azov and Black Seas.
Russia and Ukraine have set up a joint crisis centre to deal with the situation, but aircraft have been left grounded by bad weather.