A few dozen civilians managed to leave the strategically important southeastern port in a small bus convoy on Wednesday, according to Reuters witnesses, escaping the fiercest battle of the nearly eight-week-long war.
A Ukrainian marine commander, Serhiy Volny, said fighters there may not be able to hold out for much longer. President Volodymyr Zelenskiy has said an estimated 1000 civilians are also sheltering at the steel works.
Ukraine is ready for a "special round of negotiations" with no conditions "to save our guys, (the far right) Azov (battalion), military, civilians, children, the living and the wounded," negotiator Mykhailo Podolyak tweeted.
Fighters remain holed up in the plant and have ignored an ultimatum by Russia to surrender. David Arakhamia, a second negotiator, said in an online post that he and Podolyak were in constant contact with Ukrainian forces in the city.
"Today, in a conversation with the city defenders, a proposal was put forward to hold direct negotiations, on site, on the evacuation of our military garrison," he said.
"For our part, we are ready to arrive for such negotiations at any time as soon as we receive confirmation from the Russian side."
Fewer civilians than hoped left on Wednesday, according to the authorities.
More than five million Ukrainians have fled abroad since Russia invaded on February 24, the biggest attack on a European state since 1945. Russia has yet to capture any major cities.
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.
On Wednesday, Russia conducted a first test launch of its Sarmat intercontinental ballistic missile, a new and long-awaited addition to its nuclear arsenal.
"This truly unique weapon will ... provide food for thought for those who, in the heat of frenzied aggressive rhetoric, try to threaten our country," President Vladimir Putin said.
Russia calls its incursion a "special military operation" to demilitarise and "denazify" Ukraine. Kyiv and its Western allies reject that as a false pretext.
The West has responded with a slew of sanctions.
The United States on Wednesday imposed restrictions on dozens of people and entities, including a commercial bank and a virtual currency mining company, targeting the evasion of existing rules.
Mariupol, once a prosperous city of 400,000, is now a wasteland where corpses lie in the streets and Russia was hitting the Azovstal steel plant with bunker-buster bombs, the government in Kyiv said.
Large black clouds of smoke billowed from there on Wednesday as evacuees queued to get onto buses, some clutching luggage or carrying small pets in their arms.
Ukraine's Deputy Prime Minister Iryna Vereshchuk accused Russian forces of failing to observe a local ceasefire agreement long enough to allow large numbers of people to leave.
Russia did not immediately respond to the allegation.
It denies targeting civilians and has blamed Ukraine for the failure of earlier attempts to organise humanitarian corridors out of Mariupol.
Just five people in Mariupol had surrendered following Moscow's surrender-or-die ultimatum, Russian-backed separatists said.
Ukraine has proposed swapping Russian prisoners of war in exchange for safe passage for trapped civilians and soldiers.
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.