Georgia and Russia have been trading accusations in recent days
Georgia has accused Russia of trying to seize more of its territory as the anniversary of last year's brief war between the two countries approaches.
Tbilisi said Russian troops had been moving border posts along the boundary between Georgia and its Russian-backed breakaway region of South Ossetia.
Georgia's foreign ministry called Sunday's moves "extremely alarming".
Russia has accused Georgia of firing mortars into South Ossetia. EU monitors said they could confirm neither claim.
Russian troops had entered the village of Kveshi near South Ossetia and put up new border posts, the Georgian foreign ministry said in a statement on Monday, condemning the alleged move as a deliberate provocation.
The statement said the incident represented "an attempt by the Russian occupants to penetrate into the depth of Georgian territory".
On Saturday, Moscow accused Georgia of targeting South Ossetia with mortar fire and said it would respond if this continued.
The EU Monitoring Mission in Georgia (EUMM), the only organisation now monitoring the border, said there was currently no evidence for either accusation.
The EUMM had intensified patrols around the border area and called on both sides to refrain from making accusations to escalate tensions at such a sensitive time, a spokesman told the BBC.
Last year's conflict erupted on 7 August as Georgia tried to retake control of South Ossetia. Russia quickly repelled the assault, and built up its military presence in both South Ossetia and Georgia's other breakaway region of Abkhazia.
Analysts say both sides are using the approaching anniversary to try and score political points against each other.
Georgian President Mikheil Saakashvili has faced mass protests calling for his resignation since April.
The opposition accuse the Georgian president of bungling the 2008 war, and failing to strengthen the rule of law and improving democratic freedoms.
Critics say Mr Saakashvili has not addressed how he plans to regain control of South Ossetia and Abkhazia, which are both now recognised and supported as independent states by Russia.
US Vice-President Joe Biden made clear on a visit to Georgia last month that the US would not support any attempt to solve the impasse militarily.
Under a new official agreement with South Ossetia, Russia is now in charge of the territory's border protection.