Russian-backed rebels in Abkhazia say they have begun an operation against Georgian forces, which now appear under pressure on two fronts.
The rebels say they are trying to push Georgian forces from a strategic gorge in the west of the breakaway province, BBC reported.
Elsewhere, Georgian troops have withdrawn to defend the capital, Tbilisi, against any Russian assault.
France's president is visiting Russia and Georgia on Tuesday, despite Russian criticism of a new draft UN resolution.
Meanwhile the US president has strongly attacked what he called Russia's "invasion".
George W Bush said Russian actions in Abkhazia and the other breakaway province of South Ossetia were "unacceptable in the 21st Century" and that Moscow was guilty of a "dramatic and brutal escalation".