Video of the incident shows a Russian vessel in the distance shadowing the carrier and its destroyer escort.
Footage was shared by ITV as a team of news reporters visited the US warship.
The vessel is the US’s main weapon against ISIS as it launches strikes on their positions in Syria.
War tensions have flared however after the US, France and Britain launched airstrikes against Vladimir Putin’s pal Bashar al-Assad.
USS Harry S. Truman – the 116,000-ton vessel – is the flagship of the US Navy’s mighty Carrier Strike Group 8.
The warship set sail from the US last month along with a destroyer squadron to become US President Donald Trump’s leading force in the Middle East.
It has been joined by guided-missile destroyers USS Arleigh Burke, USS Bulkeley, USS Forrest Sherman, USS Jason Dunham and USS The Sullivans and USS Farragut.
Russian warships have also been filing into the Med from the Black Sea as Putin gives his full backing to Assad.
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