EU monitors have not been allowed into S Ossetia to verify reports
Russian forces have been put on a higher state of readiness ahead of this week's first anniversary of the war with Georgia, Russia says.
The foreign ministry accused Georgia of a series of "provocations".
Russia claims villages in Georgia's breakaway region of South Ossetia have been hit by mortar fire.
Georgia, meanwhile, says Russia has moved border posts further onto its territory. EU monitors on the ground say neither claim can be confirmed.
The EU has urged both sides to show restraint as the anniversary of the war, on 7 August, approaches.
The EU Monitoring Mission in Georgia (EUMM) is the only organisation now monitoring the boundary line.
It has called for unrestricted access on both sides of the boundary. Russian and Ossetian forces have not allowed EU personnel into South Ossetia so far, to check the allegations of Georgian attacks.
"The situation is very worrying and the Georgian provocations ahead of the anniversary of last year's war are not halting," Russian foreign ministry spokesman Andrei Nesterenko said in a statement.
"In connection with this, the battle-readiness of Russian troops and border guards stationed in South Ossetia has been heightened," he said.
"At the moment, the main thing is not to allow an escalation and development of the shootings into a more serious clash. We are doing and will do everything to avoid this."
However, Russia has threatened "to respond" if alleged mortar attacks continue.
Last year's conflict erupted on 7 August as Georgia tried to retake control of South Ossetia. Russia quickly repelled the assault and pushed its forces deeper inside Georgia, before pulling back.
Since the war Russia has built up its military presence in both South Ossetia and Georgia's other breakaway region of Abkhazia, and has recognised both as independent.
Analysts say both sides are using the approaching anniversary to try and score political points against each other.