A firefighter walks outside a destroyed apartment building after a bombing in Kyiv, Ukraine, on Tuesday. AP

Russian forces are pounding Ukrainian cities and edging closer to the capital, Kyiv, in a relentless bombardment that keeps deepening the humanitarian crisis in this war, now in its third week.

Local authorities said on Tuesday's bombardments on Kyiv killed at least five people as buildings were set ablaze and people were buried under rubble. Russia denies targeting civilians and launched new assaults on the port city of Mariupol, making bloody advances on the ground.

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Ukrainian President Volodymyr Zelenskyy suggested there was still some reason to be optimistic negotiations might yet yield an agreement with the Russian government.

Ukrainian servicemen carry a man injured during a shelling attack into hospital number 3 in Mariupol. AP

He said peace talks which resume on Wednesday were sounding more realistic but more time was needed, after Russian air strikes killed five people in the capital Kyiv and the refugee tally from Moscow's invasion reached 3 million.

Earlier, Interior Ministry adviser Anton Gerashchenko said Russian warships around midnight fired missiles and artillery at the Ukrainian sea coast near Tuzla, to the south of Odesa.

"They fired a huge amount of ammunition from a great distance,” he said on Facebook.

Gerashchenko said Russia wanted to test Ukraine’s coastal defence system.

A firefighter comforts a woman outside a destroyed apartment building after a bombing in Kyiv. AP

He said there was no attempt to land troops. He didn’t say whether any of the shelling hit anything.

After their delegations met Tuesday via video, Zelenskyy said Russia’s demands were becoming "more realistic.” The sides were expected to speak again later Wednesday.

"Efforts are still needed, patience is needed,” he said in his video address to the nation. "Any war ends with an agreement.”

Developments on the diplomatic front and on the ground occurred as the number of people fleeing Ukraine amid Europe’s heaviest fighting since World War II eclipsed 3 million.

A woman pushes a trolley with her belongings during the Ukraine-Russia conflict in the Donetsk region. Reuters

Zelenskyy, previewing his speech to the US Congress, thanked President Joe Biden and "all the friends of Ukraine” for $13.6 billion in new support.

He appealed for more weapons and more sanctions to punish Russia and repeated his call to "close the skies over Ukraine to Russian missiles and planes.”

He said Russian forces on Tuesday had been unable to move deeper into Ukrainian territory but had continued their heavy shelling of cities.

Over the past day, 28,893 civilians were able to flee the fighting through nine humanitarian corridors, although the Russians refused to allow aid into Mariupol, he said.