Chechen warlord Ramzan Kadyrov, said to be personally leading the devastating blitz on Mariupol, has said the city will fall on Thursday.
The top ally of Vladimir Putin said Russian forces will seize the last main stronghold of resistance in the besieged city today after Ukraine proposed talks on evacuating troops and civilians there.
Mariupol would be the biggest city to be seized by Russia since invading Ukraine eight weeks ago in an attack that has taken longer than some military analysts expected, seen over five million people flee abroad and turned cities to rubble.
The last defence in the city is said to be centred on the Azovstal steel plant, along with about 1,000 civilians who are in hiding.
Kadyrov, the head of Russia's republic of Chechnya, whose forces have been fighting in Ukraine, said: "Before lunchtime, or after lunch, Azovstal will be completely under the control of the forces of the Russian Federation."
A few dozen civilians managed to leave the strategically important southeastern port on Wednesday in a small bus convoy, escaping the fiercest battle of the war.
A Ukrainian marine commander, Serhiy Volny, said fighters at the steel works may not be able to hold out for much longer.
President Volodymyr Zelenskiy has said an estimated 1,000 civilians are sheltering there.
Mariupol, once a prosperous seaside city of 400,000, is now a wasteland where corpses lie in the streets as Russia slams the Azovstal steel plant with bunker-buster bombs, the government in Kyiv said.
The deputy commander of the Azov Regiment in Mariupol, Svyatoslav Kalamar, said several bunkers under the plant still held about 80-100 civilians each.
Black smoke billowed from the plant on Wednesday as evacuees queued to get onto buses.
Pensioner Tamara, 64, said she was going to stay with her sister in Zaporizhzhia. She was leaving with her husband, daughter, son-in law and grandson.
"It is a pleasure ... to leave after this nightmare. We lived in basements for 30 days," she said tearfully.
Ukraine accused Russian forces of failing to observe a local ceasefire agreement long enough to allow large numbers of people to leave.
Moscow denies targeting civilians and has blamed Ukraine for the failure of earlier attempts to organise humanitarian corridors out of Mariupol.
If Russia captured Mariupol it would link territory held by pro-Russian separatists in the east with the Crimea region that Moscow annexed in 2014.
Kyiv has proposed swapping Russian prisoners of war for safe passage for trapped civilians and soldiers. It was not known if Russia had responded to the offer of special negotiations.
Fighters remain holed up in the plant and have ignored an ultimatum by Russia to surrender. Fewer civilians than hoped left on Wednesday.
Ukraine said it had so far held off an assault by thousands of Russian troops attempting to advance in what Kyiv calls the Battle of the Donbas, a new campaign to seize two eastern provinces Moscow claims on behalf of separatists.
Russia's forces had carried out strikes on dozens of military facilities in eastern Ukraine and had shot down a Ukrainian Mi-8 helicopter near the village of Koroviy Yar, its defence ministry said.
Meanwhile, Putin has said a first test launch on Wednesday of Russia's Sarmat intercontinental ballistic missile, a new and long-awaited addition to its nuclear arsenal, would "provide food for thought for those who, in the heat of frenzied aggressive rhetoric, try to threaten our country."
Russia calls its incursion a "special military operation" to demilitarise and "de-nazify" Ukraine.
Kyiv and its Western allies reject that as a false pretext for a war of choice.
