"Over the past week, several thousand Mariupol residents were deported onto the Russian territory," the city council said in a statement on its Telegram channel late on Saturday.
Russian news agencies have said buses carried several hundred people Moscow calls refugees from Mariupol to Russia in recent days.
The council also said Russian forces bombed a Mariupol art school on Saturday in which 400 residents had taken shelter, but the number of casualties was not yet known.
Reuters could not independently verify the claims.
Russia denies targeting civilians.
Many of Mariupol's 400,000 residents have been trapped for more than two weeks as Russia seeks to take control of the city, which would help secure a land corridor to the Crimea peninsula that Moscow annexed from Ukraine in 2014.
President Vladimir Putin calls the assault on Ukraine, which began on February 24, a "special operation" aimed at demilitarising the country and rooting out people he terms dangerous nationalists.
Western nations call it an aggressive war of choice and have imposed punishing sanctions on Russia aimed at crippling its economy.
The Mariupol bombardment has left buildings in rubble and severed central supplies of electricity, heating and water, according to local authorities.
Rescue workers were still searching for survivors in a Mariupol theatre that local authorities say was flattened by Russian air strikes on Wednesday. Russia denies hitting the theatre.
Zelenskiy said the siege of Mariupol was a war crime. "To do this to a peaceful city ... is a terror that will be remembered for centuries to come," he said in a late night broadcast.
Still, he said, peace talks with Russia were needed although they were "not easy and pleasant".
Air raid sirens sounded across Ukrainian cities on Sunday and Russia's defence ministry said cruise missiles were launched from ships in the Black Sea and Caspian Sea, as well as hypersonic missiles from Crimean airspace.
The hypersonic missiles travel faster than five times the speed of sound and their speed, manoeuvrability and altitude make them difficult to track and intercept.
They were deployed by Russia for the first time in Ukraine on Saturday, according to Russia's Interfax news agency, in a strike which Moscow said destroyed a large underground depot for missiles and aircraft ammunition.
A spokesperson for the Ukrainian Air Force Command confirmed the attack in the western Ivano-Frankivsk region, but said the Ukrainian side had no information on the type of missiles used.
Ukraine's Deputy Prime Minister Iryna Vereshchuk said seven humanitarian corridors would open on Sunday to enable civilians to leave frontline areas. Ukraine has so far evacuated a total of 190,000 people from such areas, Vereshchuk said on Saturday.
The UN human rights office said at least 847 civilians had been killed in Ukraine as of Friday, although it says the real toll is thought to be considerably higher since its monitoring team has not yet been able to verify casualty reports from several badly hit cities.
The Ukrainian prosecutor general's office said 112 children have been killed.
Russian forces have taken heavy losses since the start of the invasion. Long columns of troops that bore down on the capital Kyiv have been halted in the suburbs.
Zelenskiy's office said on Sunday Ukraine sees a high risk of an attack launched from Belarus on the Volyn region, which lies to the north of the western city of Lviv.
