Abkhaz separatist fighters could be seen in Sukhumi on Monday
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.
Elsewhere, Georgian troops have withdrawn to defend the capital, Tbilisi, against any Russian assault.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".
The BBC's Gabriel Gatehouse, between the town of Gori and the Georgian capital, says many people in the region now believe that Russian forces intend to push on south towards Tbilisi.
He reports seeing explosions around Gori during Tuesday morning, but says the main road into and out of the town is now open again following a hasty evacuation of Georgian troops and civilians on Monday afternoon.
However, other reports say that Russian troops now control many key bridges and roads across Georgia, leaving the capital isolated.
Many Tbilisi residents have been stocking up with food and fuel, and correspondents say there have been clear signs of panic and confusion in the city.
The Russian-backed separatists' government in Abkhazia said its forces aimed to "squeeze" Georgian troops out of the upper part of the Kodori Gorge.
They launched their attack at 0600 local time (0200 GMT) and Russian TV has reported heavy gunfire and air strikes by what it says are Abkhaz planes.
There was no immediate confirmation of the offensive from the Georgian government.
Our visits to these hospitals confirm that local medical facilities are dealing with a large number of wounded and dead
Abkhazia, a much bigger province than South Ossetia, also broke away from Georgia during the collapse of the USSR in the early 1990s.
Russian troops who arrived there in recent days launched raids into Georgia proper on Monday, destroying a military base in the town of Senaki and taking control of another town, Zugdidi.
Georgia has been withdrawing its troops and armour towards Tbilisi after four days of bloody fighting in South Ossetia with Russian troops and rebel fighters.
The Red Cross has said it is still too early to say how many people have been killed or injured by the fighting, which has created thousands of refugees on both sides.
French President Nicolas Sarkozy, whose country currently chairs the EU, is to meet Russian President Dmitry Medvedev in Moscow before travelling to Georgia for talks with its president, Mikhail Saakashvili.
Broke away from Georgia in 1992-1993 war
De-facto independence not recognised internationally
2,000 Russian troops there sent as peacekeepers
Georgia seized strategic Kodori Gorge in Abkhazia in 2006
Abkhazia rejected Georgian offer of autonomy within federal state
However, Russia's envoy to the UN, Vitaly Churkin, has already dismissed a three-point French peace plan put to the Security Council in a draft resolution.
Ambassador Churkin said the plan contained "no reference to Georgian aggression or Georgian atrocities".
"What we have seen happening in the past 24 hours is that the Georgian military activity in South Ossetia is continuing... We see continued general mobilisation of the Georgian forces," he said.
"I cannot see us accepting this French draft."
Moscow has called for an emergency meeting with Nato to discuss the conflict, which erupted after the Georgians launched a surprise attack on Thursday night to retake South Ossetia and Russia sent in its forces to eject them.
In his comments on Monday, President Bush strongly criticised Russia, suggesting it might be planning to depose the Georgian government.
"Russia's government must respect Georgia's territorial integrity and sovereignty," he said.
"The Russian government must reverse the course it appears to be on."
But Mr Bush stopped short of specifying any consequences if Russia failed to heed Western appeals for an end to the fighting.
Whatever the unease in many Western capitals at the rash thinking that led President Saakashvili to order his troops into South Ossetia, there is growing outrage, BBC diplomatic correspondent Jonathan Marcus reports.
Mikhail Saakashvili on the escalating conflict with Russia
Many believe that Moscow has gone too far, he says.
By moving beyond the territory of the separatist enclaves, Russia's forces are close to cutting key transport links between Tbilisi and Georgia's Black sea ports, raising concerns about its wider strategic goals.
President Sarkozy's diplomatic round on Tuesday is a tangible signal of European concern, our correspondent says.
In Moscow, he may have uncompromising words for President Dmitry Medvedev but in Tbilisi he is likely to be hailed in much the same way as former French President Francois Mitterrand was feted when he visited Sarajevo during the dark days of the siege in 1992, he adds.
But the harsh truth is that the West has relatively few diplomatic cards to play against a powerful neighbour which controls much of Europe's energy supplies, he says.
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