"Sparta II" belongs to the company "Oboronlogistika" controlled by the Ministry of Defense of Russia.
It is under US sanctions.
In July, the merchant ship passed the Bosphorus unhindered on the way from the Syrian port of Tartus and arrived in Navaras, carrying at least 11 pieces of military equipment, the agency claims.
According to the latest data from the MarineTraffic vessel tracking system, "Sparta II" is still in the port of Novorossiysk.
Bloomberg's sources claim that Russia has repeatedly used merchant vessels to transport weapons and military equipment across the Black Sea to circumvent sanctions imposed by Turkey.
The representative of the Turkish side said that the ship can be inspected only if it is suspected of violating the law.
The White House did not comment on the situation.