Russia has continued shelling parts of the city, despite renewed calls from the West for restraint.
Battles have been reported between the opposing sides in some of Grozny's outer suburbs, while Russian shells have continued to fall elsewhere in the city.
Russia has completed its seizure of the town of Shali to Grozny's east, the last rebel stronghold apart from the capital itself, which Moscow says it can take by year's end.
Moscow's Emergencies Minister is to travel to Chechnya again tomorrow to try to talk to the Chechen leadership about getting civilians out safely.
The head of Europe's security organisation, the OSCE, says safe corridors are not enough.
He has called for a ceasefire on both sides to ensure civilians get out.
More civilians have begun to use Russian safe corridors out of Grozny.
Moscow says just over 2,000 have now left, though tens of thousands are thought to be still stranded with little food and scarce water.
Refugees have begun crossing the border into neighbouring Ingushetia, saying many more are unable to get out as Russia continues its attacks on the city.
One Chechen woman crossing the border pleaded with Russian troops not to take all her belongings, but they did anyway.
Others said they left family members behind and feared they would be victims of Russia's bombs.
Rachel Denber from Human Rights Watch, who is on the border, said: "One person we talked to said there were still 30 people in the basement where he was staying and he had no idea how they were going to get out and their food supply was dwindling."
=A9 1999 Australian Broadcasting Corporation
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