Speaking at an annual press conference in the Russian capital Moscow, Lavrov said that the Daesh militant group is attempting to create terrorist cells in certain Central Asian countries as well as in Georgia's Pankisi Gorge.
The top Russian diplomat added that some detained terrorists have confessed to receiving training in the troubled region and having links with the terrorist group.
Parts of Georgia's porous border with Russia lie in remote areas of the Caucasus mountains and there have been reports in the past about militants from Chechnya and other Russia's turbulent North Caucasus being trained there.
Moscow in the past has said that Georgia's Pankisi Gorge on the border with Chechnya serves as a hideout for militants. Georgia, which has a turbulent and problematic relationship with Russia, has denied the allegations, Press TV reported.
Russia has been fighting militants since the mid-1990s in its North Caucasus region, where the republics of Chechnya, Dagestan, and Ingushetia have been the scene of sporadic attacks and militant clashes.