Iran Press/ Europe: They said the vessel's engine room was damaged in the overnight strike in the Kerch Strait. No-one was hurt.
Russia's maritime transport agency later named the vessel as a Sig tanker. It gave its location as 17 miles (27km) south of the Crimean Bridge.
The Kerch Strait connects the Black Sea and the Sea of Azov, separating Crimea - Ukraine's peninsula annexed by Moscow in 2014 - and Russia's Taman peninsula.
Russia's state-run Tass news agency quoted an official from the country's regional Marine Rescue Co-ordination Centre (MRCC) as saying that two tugs had already arrived at the scene of the attack - just to the south of the Kerch Strait.
"The engine room was damaged. Not much, but it was damaged," the official said.
Russia's maritime transport agency RosMorRechFlot later said the vessel had a hole "in the area of [the] engine room near the waterline from the starboard side, presumably as a result of an attack by a sea drone."
"The ship is afloat," it added.
Naval drones, or sea drones, are small, unmanned vessels which operate on or below the water's surface.
Russian state-run media also reported that lights on the Crimean Bridge - further north - was turned off and all traffic halted amid warnings of an imminent attack.
If the attack is confirmed, it would be the latest such drone attack in the area.
Russia's defence ministry said it had repelled a Ukrainian attack on its naval base in Novorossiysk which involved two sea drones, but did not admit any damage.
Novorossiysk, a major hub for Russian exports, lies to the south-east of the Kerch Strait.
Clashes in the sea have increased in recent weeks, after Russia abandoned a major UN deal that enabled grain to be safely exported across the Black Sea.
In a separate development, Saudi Arabia is due to host talks later on Saturday on how to end the war in Ukraine.
Invitations have been sent to about 30 countries - but not Russia - to attend the meeting in the Red Sea city of Jeddah.
Russian President Vladimir Putin started a military operation in Ukraine in February 2022.
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