Taking the Azovstal industrial district, where the marines have been holed up, would give the Russians full control of Mariupol, Ukraine's main Sea of Azov port, and reinforce a southern land corridor before an expected new offensive in the country's east.
Surrounded and bombarded by Russian troops for weeks and the focus of some of the fiercest fighting in the war, Mariupol would be the first major city to fall since Russia invaded Ukraine on 24 February.
Russia's defence ministry said 162 officers were among 1026 soldiers of the 36th Marine Brigade who had surrendered to Russian and pro-Russian separatist forces near the Illich Iron and Steel Works. Russian television showed pictures of what it said were marines giving themselves up, many of them wounded.
Ukraine's general staff said Russian forces were attacking Azovstal and the port, but a defence ministry spokesman said he had no information about any surrender.
Reuters journalists accompanying Russian-backed separatists saw flames billowing from the Azovstal area on Tuesday.
On Monday, Ukraine's 36th Marine Brigade said it was preparing for a final battle in Mariupol that would end in death or capture as its troops had run out of ammunition.
Tens of thousands of people are believed to have been killed in Mariupol and Russia has been massing thousands of troops in the area for a new assault, Ukrainian President Volodymyr Zelenskiy said.
"We have destroyed more Russian weapons and military equipment than some armies in Europe currently possess. But this is not enough," he said in an online video, adding that if Ukraine did not get more tanks, jets, and missile systems, other countries in Europe would be the next targets of Russia.
France and Germany both pledged to send more weapons.
Ukraine accuses Russia of blocking aid convoys to tens of thousands of civilians trapped in Mariupol. Its mayor, Vadym Boichenko, said Russia had brought in mobile crematoria "to get rid of evidence of war crimes" - a statement it was not possible to verify. Moscow has blamed Ukraine for civilian deaths and accused Kyiv of denigrating Russian armed forces.
Chechen leader Ramzan Kadyrov, an ardent supporter of Russian President Vladimir Putin, urged remaining Ukrainians holed up in Azovstal to surrender.
"Within Azovstal at the moment there are about 200 wounded who cannot receive any medical assistance," Kadyrov said in a Telegram post. "For them and all the rest it would be better to end this pointless resistance and go home to their families."
Moscow's incursion into Ukraine has seen more than 4.6 million people flee abroad, killed or wounded thousands and left Russia increasingly isolated on the world stage.
US President Joe Biden said for the first time that Moscow's invasion of Ukraine amounted to genocide, a term denounced by Kremlin spokesman Dmitry Peskov.
"This is hardly acceptable from a president of the United States, a country that has committed well-known crimes in recent times," Peskov told reporters.