Russian state media reported that the USS Ross was acting "aggressively".
The US Department of Defense, however, said the ship was "well within international waters at all times, performing routine operations".
The US Navy released video on Monday of a Russian plane passing as close as 500 metres to the USS Ross.
"The crew of the ship acted provocatively and aggressively, which concerned the operators of monitoring stations and ships of the Black Sea Fleet," according to a military source quoted by Russian state media.
"Su-24 attack aircraft demonstrated to the American crew readiness to harshly prevent a violation of the frontier and to defend the interests of the country."
Pentagon spokesman Col Steve Warren said on Monday that the Russian planes were not armed with weapons and the USS Ross maintained its course.
The ship's deployment to the Black Sea had been publicly announced, the Pentagon added.
Russia's Defence Ministry would not comment on the reports.
The Black Sea encounters are the latest sign of military tension between Russian and the West, amid hostilities in eastern Ukraine.
Last month, the UK and Sweden scrambled fighters to intercept Russian bombers near their borders.
Russian warplanes have made several close passes in recent days over a US destroyer sailing in the Black Sea, the Pentagon said.