Russian troops and armour pushed into two separatist regions of Georgia and Georgian forces shelled the Russian-held capital of South Ossetia as the Caucasus crisis worsened.
Russian bombers again blasted Georgian targets overnight, the Tiblisi government said.
French Foreign Minister Bernard Kouchner, in Georgia on a peace mission, called for the fighting to cease. But the Russian troop build-up indicated Moscow was brushing aside criticism by US President George Bush of a "disproportionate response".
The simmering conflict between Russia and its small, former Soviet neighbour erupted last Thursday when Georgia suddenly sent forces to retake South Ossetia, a pro-Russian province that threw off Georgian rule in the 1990s.
Moscow responded with a counter-attack that drove Georgian forces out of the devastated South Ossetian capital Tshkhinvali on Sunday. Russia says more than 2000 people have been killed in the fighting and thousands more are homeless but these figures have not been independently verified.
The crisis in the Caucasus has triggered alarm in the West Georgia is an important energy transit route, with a key pipeline carrying oil west from the Caspian to European markets.
Russia has been rankled by Georgia's pro-Western policies and its drive for NATO membership.
Russian financial markets have slid to their lowest levels in two years as investors panicked over the conflict. Oil prices rose again on Monday after a recent retreat from record levels, with crude topping $116 a barrel.
London and Washington both backed their ally Georgia.
British Foreign Secretary David Miliband condemned Moscow for bombing targets "well beyond" South Ossetia and said there was widespread concern about the escalating violence.
But Russia, angered by what it saw as an unwarranted Georgian attack on South Ossetia last week and determined to avoid Tbilisi attacking another pro-Russian separatist area, was in no mood for compromise.
Moscow issued an ultimatum to over 1500 Georgian forces stationed in the Zugdidi district near Abkhazia, a second separatist area west of South Ossetia, to disarm or face attack but Georgia swiftly rejected the demand.
Russia said it had boosted its forces in Abkhazia beyond normal levels, stationing more than 9000 paratroopers and 350 pieces of armour there to "rule out a repetition of the situation Russian peacekeepers faced in Tskhinvali".
Moscow deployed yet more armour and troops across its southern border into North Ossetia on Monday.
A Reuters reporter said T-72 tanks and Hurricane multiple rocket launchers choked the road between Russia and Tskhinvali as military convoys headed south.
Russia originally had up to 3000 troops stationed as peacekeepers in Abkhazia and South Ossetia combined under a ceasefire agreement brokered in the early 1990s, when the two regions threw off Georgian rule.
But Georgia and some Western governments say that by supporting the separatists politically and financially before last week's clashes and now extending fighting into other parts of Georgia, Moscow has now become a party to the conflict.
Bush, speaking in Beijing on his final day at the Olympics, said he had expressed grave concern to Russian Prime Minister Vladimir Putin about "unacceptable violence" in Georgia.
"I was very firm with Vladimir Putin," Bush said. "We strongly condemn bombing outside of South Ossetia."
Sounds of explosions rocked Tskhinvali, now controlled by Russian troops, on Monday morning. Soldiers said several Russian peacekeepers were killed in Georgian shelling.
Interfax news agency quoted a South Ossetian spokeswoman as saying the Georgians had fired Katyusha rockets, killing three Russian peacekeepers and wounding 18. There was no independent confirmation of the casualties.
Georgia said up to 50 Russian fighter jets attacked Georgia overnight, though this could not be independently confirmed.
"Several dozen Russian bombers are in the Georgian skies and have been attacking throughout the country over the past several hours," the foreign ministry said in a statement.
Georgia offered Moscow a ceasefire and peace talks on Sunday and said it had pulled its troops back from South Ossetia.
Russia said Georgia had not stopped military action and demanded an unconditional Georgian withdrawal.
Kouchner, whose country holds the rotating presidency in the European Union, met Georgian President Mikheil Saakashvili on a mission to try to end the war. He was expected to go to Moscow on Monday evening.
After meeting Saakashvili, Kouchner said a "controlled withdrawal of troops" was his main priority.
"Coming back to the table, negotiations, peace talks, a political solution. That's it. Easy to say, very difficult to do," Kouchner told reporters in Tbilisi.
A Georgian government source said on Sunday 130 Georgian civilians and military personnel had been killed and 1165 wounded, many because of Russian bombing inside Georgia. Russia denies hitting civilian targets.