The reported assault on the eve of Orthodox Easter came after the Kremlin claimed its military had seized all of the shattered city except for the Azovstal plant and as Russian forces pounded other cities and towns in southern and eastern Ukraine.
A three-month-old baby was among six people killed when Russia fired cruise missiles at the Black Sea port city of Odessa, officials said.
Russia's defence ministry said its forces had used high precision missiles on Saturday to destroy a logistics terminal in Odessa where a large number of weapons supplied by the US and European countries were being stored.
Zelensky, meanwhile, said he would meet on Sunday in Kyiv with US Secretary of State Antony Blinken and Secretary of Defense Lloyd Austin.
He announced the visit at in a news conference and did not immediately share more detail.
The White House declined to comment.
Zelensky also bemoaned the death of the infant in Odessa.
"The war started when this baby was one month old. Can you imagine what is happening?" he said.
"They are just bastards... I don't have any other words for it, just bastards."
The fate of the Ukrainians in the sprawling seaside steel mill in Mariupol was not immediately clear.
Earlier on Saturday, a video emerged reportedly taken two days earlier in which women and children holed up underground, some for as long as two months, said they longed to see the sun.
The video was released by the Azov battalion, a militia set up by Ukrainian ultra-nationalists in 2014 and later incorporated as a regiment in Ukraine's national guard, which has played a prominent role in the defence of Mariupol.
"We want to see peaceful skies, we want to breathe in fresh air," one woman in the video said.
"You have simply no idea what it means for us to simply eat, drink some sweetened tea. For us, it is already happiness."
As the battle for the port ground on, Russia claimed it had taken control of several villages elsewhere in the eastern Donbas region and destroyed 11 Ukrainian military targets overnight, including three artillery warehouses.
Russian attacks also struck populated areas.
Associated Press journalists observed shelling in residential areas of Kharkiv, Ukraine's second-largest city, and regional Governor Oleh Sinehubov said three people were killed.
In the Luhansk area of the Donbas, Governor Serhiy Haidai said six people died during the shelling of a village, Gorskoi
In Sloviansk, a town in northern Donbas, AP witnessed two soldiers arriving at a hospital, one of them mortally wounded.
Nearby a small group of people gathered outside a church where a priest blessed them with water on Holy Saturday.
While UK officials said Russian forces had not gained significant new ground, Ukrainian officials announced a curfew ahead of Easter Sunday, a sign of the war's disruption and threat to the entire country.
Mariupol has been a key Russian objective since the invasion began on February 24 and has taken on outsized importance in the war.
Completing its capture would give Russia its biggest victory yet, after a nearly two-month siege reduced much of the city to a smoking ruin.
It would deprive the Ukrainians of a vital port, free up Russian troops to fight elsewhere and establish a land corridor to the Crimean Peninsula, which it seized in 2014.
Russia-backed separatists control parts of the Donbas.
An advisor to Ukraine's presidential office, Oleksiy Arestovich, said during a Saturday briefing that Russian forces resumed air strikes on the Azovstal plant and were trying to storm it.
A direct attempt to take the plant would represent a reversal from an order Russian President Vladimir Putin gave two days earlier.
Russian Defence Minister Sergei Shoigu reported to Putin on Thursday that the whole of Mariupol, with the exception of Azovstal, had been "liberated".
At the time, Putin ordered him not to send troops into the plant but instead to block it off.
In the attack on Odessa, Russian troops fired at least six missiles, according to Anton Gerashchenko, an adviser to Ukraine's interior minister.
Defence forces repelled some of the rockets but at least one hit, he said.