However, hundreds of fighters and civilians in dozens of bunkers were still holed up inside a huge steel works on Thursday.

Putin ordered his troops to blockade the complex “so that not even a fly” could escape.

Ukraine said Putin wanted to avoid a final clash with its forces in the city as he lacked the troops to defeat them.

In Washington DC, US President Joe Biden authorised a further $US800 million ($A1.1 billion) in military aid for Ukraine, including heavy artillery, as it faces a new Russian offensive in the east.

Ukrainian President Volodymyr Zelenskiy, in a video address to the Portuguese parliament, begged other countries to send more weapons and to impose more economic sanctions on Russia.

“This is just the first step (for Russia) to gain control of eastern Europe, to destroy democracy in Ukraine,” he said.

“Your people, who in a few days will be celebrating the anniversary of the Carnation Revolution that liberated you from dictatorship, know perfectly well what we are feeling.”

“We are fighting not only for our independence but for our survival, for our people so that they do not get killed, tortured and raped,” Zelenskiy said.

He accused the Russian army of committing many atrocities in Ukraine, including in Mariupol.

Russia denies targeting civilians and rejects what Ukraine says is evidence of atrocities, saying Ukraine has staged them.

Russia calls its incursion a “special military operation” to demilitarise and “denazify” Ukraine.

Ukraine and its allies reject that as a false pretext for an illegal war of aggression.

Control of Mariupol links territory held by separatists Russia backs in eastern Ukraine’s Donbas region with Crimea, the peninsula it seized in 2014.

Ukrainian fighters remained inside the Azovstal steel complex, one of the biggest metallurgical facilities in Europe, covering 11 sq km with huge buildings, underground bunkers and tunnels.

Putin had told the defenders to lay down their weapons and surrender or die.

But in a televised meeting at the Kremlin on Thursday, Putin congratulated Defence Minister Sergei Shoigu and Russian troops for having “successfully completed the combat effort to liberate Mariupol”.

He said it was unnecessary to storm the industrial zone and ordered such action to be cancelled.

“There’s no need to climb into these catacombs and crawl underground through these industrial facilities… Block off this industrial area so that not even a fly can get through.”

The Ukrainian military said Russian forces had nevertheless tried to storm the seaport and steel plant.

While Putin claimed his first big prize since his forces were driven out of northern Ukraine last month, it falls short of the unambiguous victory Russia has sought after months of combat in a city reduced to rubble.

“They physically cannot take Azovstal, they have understood this, they have taken huge losses there,” Ukrainian presidential adviser Oleksiy Arestovych told a briefing.

“Our defenders continue to hold it.”

Zelenskiy said 120,000 civilians were still being blocked from leaving Mariupol.

Journalists who reached Mariupol during the siege found streets littered with corpses, nearly all buildings destroyed, and residents huddled freezing in cellars, venturing out to cook scraps on makeshift stoves or to bury bodies in gardens.

Ukraine estimates that tens of thousands of civilians have died in Mariupol, some buried in mass graves.

The United Nations and Red Cross say the civilian toll is at least in the thousands.

Shoigu told Putin that Russia had killed more than 4000 Ukrainian troops in its campaign to take Mariupol and that 1478 had given themselves up.

Those figures could not be verified.

Ukrainian deputy prime minister Iryna Vereshchuk said 1000 civilians and 500 wounded soldiers needed to be brought out from the plant immediately, blaming Russian forces for the failure to establish a safe corridor that she said had been agreed.

Russia says it has taken in 140,000 civilians from Mariupol.

Ukraine says some were deported by force, in what would be a war crime.