WASHINGTON/MARIUPOL/KYIV (REUTERS) - United States President Joe Biden issued one of his strongest warnings yet that Moscow is considering using chemical weapons.

Without citing evidence, Mr Biden said Russia’s false accusations that Kyiv had biological and chemical weapons illustrated that President Vladimir Putin was considering using them himself.

Mr Putin’s “back is against the wall and now he’s talking about new false flags he’s setting up including, asserting that we in America have biological as well as chemical weapons in Europe, simply not true,” Mr Biden said at a Business Roundtable event on Monday (March 21).

“They are also suggesting that Ukraine has biological and chemical weapons in Ukraine. That’s a clear sign he’s considering using both of those.” The Russian embassy in Washington did not immediately respond to a request for comment.

Ukraine said on Monday  it would not obey ultimatums from Russia after Moscow demanded it stop defending besieged Mariupol, where hundreds of thousands of civilians are suffering through Russian bombardments laying waste to their city.

Mariupol has become a focal point of Russia's assault on Ukraine, but attacks were also reported to have intensified on the country's second city Kharkiv on Monday.

The conflict has driven almost a quarter of Ukraine's 44 million people from their homes, and Germany predicted the refugee number could reach as high as 10 million in coming weeks.

Europe said Russia was using refugees as a tool and that it was prepared to take more action on top of existing sanctions to isolate Russia from global finances and trade.

Russia's military had ordered residents of Mariupol to surrender by 5am (11am, Singapore time) on Monday, saying those who did so could leave, while those who stayed would be handed to tribunals run by Moscow-backed separatists.

President Volodymyr Zelensky's government responded that it would never bow to ultimatums and said cities such as the capital Kyiv, Mariupol and Kharkiv would always defy occupation.

"There can be no question of any surrender" in Mariupol, said Ukrainian Deputy Prime Minister Iryna Vereshchuk.

Russia's invasion, now in its fourth week, has largely stalled, failing to capture any major city, but causing massive destruction to residential areas.

Mariupol, a port city on the Sea of Azov that was home to 400,000 people, has run short of food, medicine, power and water. Ukrainian Defence Minister Oleksii Reznikov said its "heroic defenders" had helped thwart Russia elsewhere.

A part of Mariupol now held by Russian forces, reached by Reuters on Sunday, was an eerie wasteland. Several bodies lay by the road, wrapped in blankets. Windows were blasted out and walls were charred black. People who came out of basements sat on benches amid the debris, bundled up in coats.

Some, though, are managing to escape.

A total of 8,057 people were safely evacuated on Monday through seven humanitarian corridors from towns and cities under fire, said Ms Vereshchuk. Among those brought to safety were 3,007 residents of Mariupol.

Russia calls the war, the biggest attack on a European state since World War Two, a "special military operation" to disarm Ukraine and protect it from "Nazis".

The West calls this a false pretext for an unprovoked war of aggression by Mr Putin.

Mr Biden told businesses to be alert for possible cyber attacks by Russia and do more to protect themselves.

"It's part of Russia's playbook," he said in a statement.

"My administration is reiterating those warnings based on evolving intelligence that the Russian government is exploring options for potential cyberattacks."

There was no immediate reply from the Kremlin to a request for comment from Reuters out of hours on Monday.

Russia has previously rejected similar allegations including that it was responsible for hacks on Ukrainian banking and government websites in February.

In response to Mr Biden last week calling Mr Putin a "war criminal," Russia's foreign ministry said on Monday it had summoned US Ambassador John Sullivan.

"Such statements from the American president, unworthy of a statesman of such high rank, put Russian-American relations on the verge of rupture," it said.

Deputy US Secretary of State Wendy Sherman, in an interview on MSNBC, said Russia's reaction "just shows how desperate President Putin is becoming."

The conflict continues to rage in a number of locations.

The eastern cities of Kharkiv, Sumy and Chernihiv have also been hard hit by Russia's tactic of pounding urban areas with artillery as its troops have done before in Syria and Chechnya.

On Monday night, a witness in Kharkiv said she saw people on the roofs of apartment buildings dropping grenades or similar ordnance onto the streets.

A second witness, outside the city, reported hearing more intense explosions than on any day since Russian troops began attacking last month.

Reuters could not immediately verify the accounts.

In Kyiv, six bodies were laid on the pavement by a shopping mall struck overnight by Russian shelling. Emergency services combed wreckage to the sound of distant artillery fire.

Firefighters put out small blazes around the building, hunting for survivors. Ukraine said at least eight people died.

Russia said the centre was being used as a weapons store.

Ukraine said there were no strategic military objects in the area. Neither report could be independently verified.

Officials imposed a day-and-a-half curfew in the capital from Monday night, citing the likelihood of more shelling.

Britain said there was heavy fighting to the north but that Ukrainian forces had fought off an advance and most Russian forces were more than 25km from the city centre.

Ukrainian officials hope that Moscow, having failed to secure a quick victory, will cut its losses and negotiate a withdrawal. Both sides hinted last week at progress in talks on a formula which would include some kind of "neutrality" for Ukraine, though details were scarce.

West of Mariupol, the governor of Ukraine's Zaporizhzhia region said shelling had hit buses evacuating civilians from front line areas and four children were wounded in separate incidents that he blamed on Russia. Reuters could not confirm the report independently. Moscow denies targeting civilians.

Mr Biden discussed Russia's "brutal tactics" in Ukraine with European leaders on Monday and Britain said they reaffirmed their commitment to support Ukraine militarily, diplomatically and economically.

But European Union foreign ministers disagreed on whether and how to include energy in sanctions, with Germany saying the bloc was too dependent on Russian oil to declare an embargo.