Ukraine's armed forces command wrote on Facebook that Russian forces were firing and performing "offensive operations" in the Azovstal area, as well as conducting air strikes on civilian infrastructure.
Serhiy Volyna, commander of Ukraine's 36th Marine brigade forces in Mariupol, said in an interview with an opposition lawmaker that was shown on YouTube on Sunday that Russia was hitting the complex with air and artillery bombardments.
"We are taking casualties, the situation is critical... we have very many wounded men, (some) are dying, it's a difficult (situation) with guns, ammunition, food, medicines... the situation is rapidly worsening," Volyna said, speaking from his location at the plant.
Konstantin Ivaschenko, the official who has been designated mayor of Mariupol by Russia but not recognised as such by Ukraine, denied that any fighting was taking place in the city in comments reported by Russian news outlet TASS on Sunday.
Reuters could not independently verify the Ukrainian or Russian the accounts.
The Azovstal steel factory is the main remaining Ukrainian stronghold in Mariupol, a city that has seen sustained bombardment since the start of the Russian invasion on Feb. 24.
Earlier on Sunday, Ukrainian presidential adviser Oleksiy Arestovych wrote on Facebook that "Russian troops are trying to finish off the defenders of Azovstal and more than 1,000 civilians who are hiding at the plant".
Russia has denied targetting civilians in what it calls a "special military operation" to demilitarise and "denazify" Ukraine.
Later on Sunday, Arestovych said in a video address released by the president's office that Ukraine was offering Russia a "special" round of negotiations to be held in Mariupol to discuss the fate of the civilians and Ukrainian troops still trapped in the city.
The negotiations would be intended to establish an immediate ceasefire in Mariupol, "multi-day" humanitarian corridors, and the freeing or swapping of Ukrainian fighters trapped in the Azovstal plant, Arestovych said.
Russian troops surrounded the Azovstal plant in early March and have gradually taken control of most of the city.
Last Thursday, Putin declared that Mariupol had been "liberated" and publicly told his defence minister to call off the storming of the Azovstal plant so as to save the lives of Russian soldiers. Putin said the plant should be "blocked off" instead.
epa09896966 A frame grab from an undated handout drone video first published by DPR militia commander Alexander Khodakovsky and made available by the Mariupol City Council shows smoke rising from the Azovstal steel plant during airstrikes in Mariupol, eastern Ukraine, 18 April (issued 19 April 2022). The Russian Defence Ministry on 19 April 2022 issued a statement calling on the Ukrainian forces in Mariupol "to cease any hostilities and lay down their arms. All who lay down their weapons are guaranteed the preservation of life." The city council on 18 April 2022 via their official Telegram channel said that at least 1,000 civilians are sheltering in the underground shelters of the metallurgical plant, and that heavy bombs were dropped on the Azovstal plant by Russian forces. EPA-EFE/MARIUPOL CITY COUNCIL HANDOUT HANDOUT EDITORIAL USE ONLY/NO SALES 