Russia and Georgia have accused each other of launching new attacks, as diplomats press for a ceasefire in the conflict over South Ossetia.
Georgia said dozens of Russian bombers were attacking targets inside its territory, including around Tbilisi.
Russia said Georgian attacks on the South Ossetian capital Tskhinvali killed three of its troops.
But an EU delegation visiting Tbilisi said Georgia’s president had signed a draft proposal for a ceasefire.
French Foreign Minister Bernard Kouchner told the BBC that Mr Saakashvili had accepted EU proposals for a ceasefire, controlled withdrawals of troops on both sides and eventual political talks.
The delegation would now go to Moscow, Mr Kouchner said, to convince President Dmitry Medvedev to back the plan.
Earlier, Mr Medvedev said Russian troops were in control of Tskhinvali and Moscow’s military push was “largely complete”.
But the head of the Organisation for Security and Co-operation in Europe, Alexander Stubb, accompanying Mr Kouchner, said he could not predict when the conflict would end, saying only that he was “optimistic” a cessation of violence could begin “in the next few days”.
Fighting over South Ossetia erupted late last week when Georgia launched an overnight assault on the territory, which has had de facto independence since the end of a war in 1992.
Russia, which supports the breakaway province, hit back, bombing targets throughout Georgia.
The latest reports of violence came despite Georgia saying on Sunday that it would observe a ceasefire. Moscow has insisted Georgian forces withdraw fully from South Ossetia before it halts operations.
From Tbilisi, Georgia said up to 50 Russian fighter jets attacked targets inside Georgia overnight, with targets including a missile base and a radar station.
Georgia said the town of Gori, close to the South Ossetian border and used as a jumping-off point for Georgia’s push into South Ossetia, also came under overnight attack.
Elsewhere in Georgia, tensions were rising in Abkhazia, another separatist region.
Reports said a Russian general issued an ultimatum to Georgian forces to pull out of Abkhazia’s Kodori Gorge or Russian would send in its troops. Earlier, reports in Moscow said 9,000 Russian troops were being deployed to Abkhazia.
On Sunday, separatist leaders in Abkhazia announced a full mobilisation in order to drive Georgian troops from part of the region, and gave them a deadline to leave.
Georgia has accused Russia of landing 4,000 more troops in Abkhazia via the Black Sea. The separatists said Georgia had deployed a similar number of soldiers south of the Abkhaz border.
Overnight, US President George W Bush was strongly critical of Russia’s military strikes against Georgia.
Speaking in Beijing, US President Bush told NBC TV that he had spoken frankly to Vladimir Putin when the pair met at the opening ceremony of the Olympic Games last week.
“I said this violence is unacceptable,” Mr Bush said, adding: “I was very firm with Vladimir Putin. Hopefully this will get resolved peacefully.”
However, in a telephone call to Georgian President Mikhail Saakashvili, the US Vice-President, Dick Cheney, said Russian aggression “must not go unanswered”.
Mr Cheney said the continuation of violence against Georgia would have serious consequences for Russia’s relations with the US, as well as the international community.
But White House officials refused to speculate on what America might do if the Russian military action continued.
The United Nations refugee agency (UNHCR) has called on the parties to the conflict to grant safe passage to civilians trying to escape the war zone.
The UNHCR estimates that between 10,000 and 20,000 people have been displaced within Georgia, including South Ossetia, while Russia has said that a further 30,000 people have fled north into the Russian province of North Ossetia.