One of the vessels split in two and sank, while another was severely damaged and ran aground in the incident on Sunday.
The Russian government said one crew member died from hypothermia while a total of 26 were saved from both tankers.
Specialists had been deployed and are "completing preparatory work to clean up the spilt fuel oil", the government said in a statement.
It did not say what the extent of the spillage was but said it was deemed to have been "partial".
The tankers were carrying around 9,200 metric tons of oil between them, Russia's state-run TASS news previously reported.
Ukraine's environmental ministry said the leak had been "significant", posed "the threat of a large-scale environmental disaster in the Black Sea" and could have "catastrophic consequences for the marine ecosystem."
It accused Russia of failing to comply with maritime safety rules.
Both vessels -- Volgoneft-212 and Volgoneft-239 -- were reported by Russian media to be more than 50 years old.
Russia has been using a shadow fleet of old tankers to ship its vital oil exports around the world in the face of Western sanctions.
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