Street fighting and bombardments raged in Mariupol, Ukrainian officials said, a day after it rejected an ultimatum from Russia to surrender.
Hundreds of thousands are believed to be trapped inside buildings, with no access to food, water, power or heat.
Russian forces and Russian-backed separatist units had taken about half of the port city, normally home to around 400,000 people, Russia's RIA news agency said, citing a separatist leader.
"There is nothing left there," Ukrainian President Volodymyr Zelenskiy said in a video address to Italy's parliament.
Mariupol Deputy Mayor Sergei Orlov told CNN the city was under a full blockade and had received no humanitarian aid.
"The city is under continuous bombing, from 50 bombs to 100 bombs Russian aircraft drops each day ... A lot of death, a lot of crying, a lot of awful war crimes," Orlov said.
Mariupol has become the focus of the war that erupted on February 24 when Russian President Vladimir Putin sent his troops over the border on what he calls a "special military operation" to demilitarise Ukraine and replace its pro-Western leadership.
The city lies on the Sea of Azov and its capture would allow Russia to link areas in the east held by pro-Russian separatists with the Crimean peninsula, annexed by Moscow in 2014.
Putin's 27-day long incursion into Ukraine has forced more than 3.5 million to flee, brought the unprecedented isolation of Russia's economy, and raised fears of wider conflict.
Western nations plan to heap more economic pressure on the Kremlin.
US President Joe Biden will join allies in applying additional sanctions and tightening existing ones during his trip to Europe this week, national security adviser Jake Sullivan said on Tuesday.
The trip will also include an announcement on joint action to enhance energy security in Europe, which is highly reliant on Russian gas, and Biden will show solidarity with Ukraine's neighbour, Poland, with a visit to Warsaw.
Having failed to seize the capital Kyiv or any other major city with a swift offensive, Russia is waging a war of attrition that has reduced some urban areas to rubble and prompted Western concern the conflict could escalate, even to a nuclear war.
Russia's security policy dictates the country would use such weapons only if its very existence were threatened, Kremlin spokesman Dmitry Peskov told CNN on Tuesday.
"If it is an existential threat for our country, then it (the nuclear arsenal) can be used," he said.
The United Nations human rights office in Geneva said on Tuesday it had recorded 953 civilian deaths and 1557 injured since the invasion. The Kremlin denies targeting civilians.
Western officials said Russian forces were stalled around Kyiv but making some progress in the south and east.
Deputy Prime Minister Iryna Vereshchuk, speaking on Ukrainian television on Tuesday, demanded the opening of a humanitarian corridor for civilians. She said at least 100,000 people wanted to leave Mariupol but could not.
"Our military are defending Mariupol heroically," Vereshchuk said, referring to Russia's earlier demand the city surrender by dawn on Monday.
"We did not accept the ultimatum. They offered capitulation under a white flag."
Kyiv accused Moscow of deporting residents of Mariupol and separatist-held areas of Ukraine to Russia. This includes the "forcible transfer" of 2389 children to Russia from the Luhansk and Donetsk regions, Prosecutor General Iryna Venediktova said.
Moscow denies forcing people to leave, saying it is taking in refugees.
In an address overnight, President Zelinskiy drew attention to the death of Boris Romanchenko, 96, who survived three Nazi concentration camps during World War Two but was killed when his apartment block in besieged Kharkiv was shelled last week.
In killing Romanchenko, "Putin managed to 'accomplish' what even Hitler couldn't", Ukraine's Defence Ministry said.