Tass quoted the Russian headquarters in the region as saying by 9.30 a.m. troops advancing from two directions after a night of heavy artillery and aerial bombardment had reached the city center.
It said the military estimated the operation to capture the city might take three to four days. The Tass report could not be independently confirmed and officials were not immediately available for comment.
An earlier drive to seize the city, also scheduled to take several days, ran into trouble in the New Year when troops encountered fierce resistance from the entrenched rebels.
The rebel web site (www.kavkaz.org) said troops, spearheaded by tanks, were suffering heavy casualties as they battled their way into Grozny sometimes in hand-to-hand fighting.
Tass said fierce fighting was raging elsewhere in the city where up to 40,000 civilians remained trapped in cold, dank cellars with little or no food.
Refugees fleeing Grozny on Monday spoke of a city full of wounded and hungry people too exhausted or frightened to flee.
''They aren't killing any bandits,'' Zura, 49, told Reuters after reaching Chechnya's border with Ingushetia.
''They're killing old men, women and children. And they keep on bombing -- day and night. Do you know what difficulties we had getting out of the city? We had shells falling around us.''
The plight of refugees and civilians is expected to feature high on the agenda during a visit by the Council of Europe, which promotes human rights and democracy.
A delegation was due to visit Russian-controlled villages in Chechnya as well as the adjacent regions of Dagestan and Ingushetia, temporary home to some 200,000 Chechen refugees.
Tass said Russian Foreign Minister Igor Ivanov was due to fly to the organization's headquarters in Strasbourg, France, to forestall possible sanctions against Moscow on Chechnya at an emergency meeting of the Council of Europe on January 27.
Moscow Ready To Talk To The West On Chechnya
Clearly anxious to improve Russia's battered international image, Acting President Vladimir Putin devoted three hours on Monday to the visiting mission, which is led by the head of the Council's parliamentary assembly, Lord Russell-Johnston.
Putin, tipped to become Russia's next president after a March 26 election, appealed for Western understanding and said he would accept an ''international presence'' in the region to help ensure balanced foreign media coverage of the conflict.
Western criticism of the campaign has focused on the suffering of civilians, both those trapped in Grozny and elsewhere inside Chechnya and the refugees outside the province.
- Reuters - Thomson Reuters Foundation
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