The Russian Navy has been following the USS Carney since it entered the Black Sea, the National Defence Management centre reported.
The MoD's statement says that the Black Sea Fleet's "Smetlivy" (Sharp-witted) will directly monitor the warship's actions. It will also be monitored by electronic and technical means of the fleet.
According to the centre, the US destroyer entered the Black Sea at 19:20 Moscow time (16:20 GMT) on 29 June.
"In accordance with the provisions of the Montreux international convention regarding passage via the Turkish straits for warships of non-Black Sea states, the time spent by the USS destroyer in the Black Sea should not exceed 21 days", the statement said.
This is reportedly the fifth time an American warship has entered the Black Sea this year.
The 19th edition of the exercise, which has land, air and sea components, will take place in the Black Sea from 1-12 July. Nineteen nations, 32 ships, 24 aircraft and over 3,000 troops will participate. The United States, co-host of the drills, has sent the USS Carney destroyer.
Russia has repeatedly accused the US-led bloc of illegally expanding eastward in Europe in violation of its promises. NATO has been building up its military on Russia’s eastern flank, stationing troops in Estonia, Latvia, Lithuania, and Poland in 2017, while US warships have been routinely calling on Black Sea ports.