The Kremlin’s defence ministry spokesman Maj Gen Igor Konashenkov said two navy destroyers detected a Walrus-class submarine on Wednesday, which was 11 miles away from the Admiral Kuznetsow aircraft carrier.
The submarine is said to have followed the Russian aircraft carrier for more than an hour, before Moscow’s warships forced it to leave.
A Russian defence ministry spokesman said: “The vessels followed its manoeuvres for more than an hour and forced it to leave the deployment area of the aircraft carrier’s group.
“These clumsy attempts to carry out dangerous manoeuvres in the immediate proximity of a group of Russian vessels could have had grave navigational consequences.”
The spokesperson branded Holland’s decision to trail its warships as “dangerous” and warned of “grave consequences”.
In a tweet, the Dutch defence ministry said it does not comment on operations conducted by its submarines.
Russian warships have been monitored in recent weeks by Nato, with a Nato official stating that the coalition has been tracking the fleet in a “measured and responsible way, as is customary.”
Moscow’s helicopters had previously spotted several Nato submarines trailing Russian warships, including a US Virginia-class nuclear submarine.
Nato is currently preparing a 300,000-strong army, capable of being deployed within three months, in response to a mounting threat from Russia towards the West.
The alliance is also responding to an increase in espionage, hybrid warfare and cyberattacks by Russia and other non-Nato states.
Jens Stoltenberg, Secretary General of Nato, said: “We have seen Russia being much more active in many different ways.
“We have seen a more assertive Russia implementing a substantial military build-up over many years; tripling defence spending since 2000 in real terms; developing new military capabilities; exercising their forces and using military force against neighbours.
In recent months, the Kremlin has bolstered its naval presence in the Mediterranean as part of its intervention in Syria’s five year-long civil war.
Russia has been carrying out airstrikes on Syria in what it claims is a campaign to wipe out ISIS, although critics have accused the Kremlin of bombing rebel groups in an effort to prop up ally Bashar al-Assad.