The capture of 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, reinforce a southern land corridor and expand its occupation of the country's east.
Surrounded and bombarded by Russian troops for weeks and the focus of some of the heaviest fighting in the war, Mariupol would be the first major city to fall since Russia invaded Ukraine on February 24.
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 in Azovstal.
Russian television showed pictures of what it said were marines giving themselves up, many of them wounded.
The defence ministry later said Mariupol's trade seaport was under full control and remnants of Ukrainian forces were blocked and unable to escape, Interfax news agency reported.
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, a day after Ukraine's 36th Marine Brigade said its troops had run out of ammunition.
Chechen leader Ramzan Kadyrov, an ardent supporter of Russian President Vladimir Putin, urged remaining Ukrainians trapped 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."
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.
The United States, France and Germany all pledged to send more weapons, with US President Joe Biden listing artillery systems, armoured personnel carriers and helicopters.
Ukraine accuses Russia of blocking aid convoys to civilians marooned in Mariupol. Its mayor, Vadym Boichenko, said Russia had brought in mobile crematoria "to get rid of evidence of war crimes" - a statement that was not possible to verify.
On the wall of a house that had been occupied by Russian troops in a village near Kyiv, a message left for the Ukrainians read: "We did not want this ... forgive us."
The presidents of Poland, Lithuania, Latvia and Estonia became the latest in a string of European politicians to visit Kyiv but Germany's president did not join them as he had planned. Zelenskiy denied a newspaper report he had rejected the visit.
The Kremlin denounced Biden's description of Moscow's actions in Ukraine as amounting to genocide, with spokesman Dmitry Peskov saying this was unacceptable coming from the leader of a country he said had committed crimes of its own.