Russia's Black Sea Fleet forces "began to monitor" the actions of the French navy's multi-mission frigate Auvergne after it entered the Black Sea, the defence ministry said in a statement carried by Russian news agencies.

The Russian defence ministry did not offer further details.

Tensions between Moscow and the West are on the rise over Russian troops movements near ex-Soviet Ukraine.

Kiev and its allies accuse Russia of planning a possible winter invasion, while Moscow says the West is stoking tensions.

Last Thursday, Russia scrambled three of its jets to escort five French and US military aircraft over the Black Sea and away from its border, the military said.

That incident followed Russia dispatching planes to escort three French military flying near its borders in the Black Sea a day earlier.

Western countries have issued repeated warnings to Moscow against invading their ally Ukraine.

The European Union on Monday signalled its readiness to impose new sanctions targeting Russia's economy if the troop build-up leads to direct military action.

Last week, France warned Russia of "strategic and massive consequences" if Moscow attacked Ukraine.

Moscow accuses NATO member countries including France of provoking tensions in the Black Sea and says it is concerned over its security.

Ukraine has been locked in a simmering conflict with pro-Moscow separatists since 2014, when they seized two regions shortly after Russia annexed the Crimean peninsula.

The West accuses Russia of sending troops and weapons across the border to support the separatists in a conflict that has taken over 13,000 lives. Moscow denies the claims.