Mariupol, a port city on the Azov Sea, has been under siege and bombardment, with no food, medicine, power or fresh water, since the early days of Russia's invasion on February 24.
Video footage of the city shows it has been devastated.
Ukraine described the situation in Mariupol on Monday as "very difficult" and said it had been unable to establish a new safe corridor to evacuate civilians from the besieged city after it defied a Russian ultimatum to surrender.
Russia's military had ordered Ukrainians inside Mariupol to surrender by 5 A.M., saying that those who did so would be permitted to leave through safe corridors.
"Of course we rejected these proposals," Deputy Prime Minister Iryna Vereshchuk said.
She said agreement had been reached with Russia on creating eight humanitarian corridors to evacuate civilians from besieged towns and cities on Monday but Mariupol was not among them.
Efforts to reach the city with humanitarian supplies continue to fail, she said. "The situation there is very difficult," she added.
She gave no further details. Ukraine has accused Russian forces of bombarding buildings including hospitals and a theatre where it said people were sheltering last week.
Russia denies targeting civilians. Russia's Defence Ministry has blamed "Ukrainian nationalists" for the situation in Mariupol. A Russian-backed separatist leader in eastern Ukraine said it would take more than a week to take control of Mariupol.
"I am not so optimistic that two or three days or even a week will close the issue. Unfortunately, no, the city is big," the Russian news agency Interfax quoted Denis Pushilin, head of the self-proclaimed Donetsk People's Republic, as saying.
Defence Minister Oleksii Reznikov said the city's "heroic defenders" had helped prevent Russia marching on other big cities and saved many lives.
Greece's consul general in Mariupol, Manolis Androulakis, who arrived home on Sunday after escaping the siege, said: "What I saw, I hope no one will ever see." He described Mariupol as standing alongside Guernica, Leningrad and Russia's previous targets Grozny and Aleppo.