Kremlin officials have said safe corridors to allow Ukrainian civilians to escape the Russian onslaught could open on Tuesday.
Ukrainian leaders are sceptical, after prior efforts to establish evacuation routes crumbled amid renewed attacks from Russian forces.
With the invasion well into its second week, Russian troops have made significant advances in southern Ukraine but stalled in some other regions.
Shells and rockets have continued to fall on other population centres, including the Kyiv suburb of Bucha, where the mayor reported heavy artillery fire.
“We can’t even gather up the bodies because the shelling from heavy weapons doesn’t stop day or night,” mayor Anatol Fedoruk said.
“Dogs are pulling apart the bodies on the city streets. It’s a nightmare.”
In the encircled southern port of Mariupol, an estimated 200,000 people – nearly half the population of 430,000 – were hoping to flee, and Red Cross officials are waiting to hear when a corridor would be established.
Russia’s coordination centre for humanitarian efforts in Ukraine said Russia would begin a ceasefire at 10am Moscow time (7am GMT) to allow civilians to flee through special corridors agreed upon with Ukrainian authorities, according to Russian media.
Most of those corridors would lead to Russia, either directly or through Belarus, though people in Kharkiv would be allowed to travel to western Ukraine, the centre said.
The office of Ukrainian President Volodymyr Zelenskyy would not comment on the latest Russian proposal, saying only that Moscow’s plans can be believed only if a safe evacuation begins.
Demands for effective passageways have surged amid intensifying shelling by Russian forces. The steady bombardments, including in some of Ukraine’s most populated regions, have yielded a humanitarian crisis of diminishing food, water and medical supplies.
Through it all, Mr Zelenskky said Ukrainian forces are showing unprecedented courage.
“The problem is that for one soldier of Ukraine, we have 10 Russian soldiers, and for one Ukrainian tank, we have 50 Russian tanks,” Mr Zelenskyy told ABC News.
But he noted that the gap in strength was closing and that even if Russian forces “come into all our cities”, they will be met with an insurgency.
A top US official said multiple countries are discussing whether to provide the warplanes that Mr Zelensky has been pleading for.
Mariupol is short on water, food and power, and mobile phone networks are down.
Hospitals in Mariupol are facing severe shortages of antibiotics and painkillers, and doctors have performed some emergency procedures without them.
The lack of phone service left anxious citizens approaching strangers to ask if they knew relatives living in other parts of the city, and whether they were safe.
Several hundred miles west of Mariupol, Russian forces continued their offensive in Mykolaiv, opening fire on the Black Sea shipbuilding centre of a half a million people, according to Ukraine’s military.
Rescuers said they were putting out fires caused by rocket attacks in residential areas.
Ukraine’s general staff of the armed forces said in a statement that Ukrainian forces are continuing defence operations in the suburbs of the city.
The general staff said “demoralised” Russian forces are engaging in looting in places they have occupied, commandeering civilian buildings like farm buildings for military equipment, and are setting up firing positions in populated areas. The claims could not be independently verified.
In Kyiv, soldiers and volunteers have built hundreds of checkpoints to protect the city of nearly four million, often using sandbags, stacked tyres and barbed wire.
Some barricades looked significant, with heavy concrete slabs and sandbags piled more than two storeys high, while others appeared more haphazard, with hundreds of books used to weigh down stacks of tyres.
“Every house, every street, every checkpoint, we will fight to the death if necessary,” said mayor Vitali Klitschko.
In Kharkiv, Ukraine’s second-largest city, with 1.4 million people, heavy shelling slammed into apartment buildings.
In the small town of Horenka, where shelling reduced one area to ashes and shards of glass, rescuers and residents picked through the ruins as chickens pecked around them.
“What are they doing?” rescue worker Vasyl Oksak asked of the Russian attackers. “There were two little kids and two elderly people living here. Come in and see what they have done.”
At The Hague, Ukraine pleaded with the International Court of Justice to order a halt to Russia’s invasion, saying Moscow is committing widespread war crimes.
Russia snubbed the court proceedings, leaving its seats in the Great Hall of Justice empty.
Efforts to set up safe passage for civilians over the weekend fell apart amid continued Russian shelling.
Russia’s plan aimed to allow civilians to leave Kyiv, Mariupol, Kharkiv and Sumy, using routes toward Russia or its ally Belarus, which was a launchpad for the invasion.
Ukraine instead proposed eight routes allowing civilians to travel to western regions of the country where there is no shelling.
The UN humanitarian chief, Undersecretary-General Martin Griffiths, addressed the Security Council and urged safe passage for people to go “in the direction they choose”.
The battle for Mariupol is crucial because its capture could allow Moscow to establish a land corridor to Crimea, which Russia seized from Ukraine in 2014.
The fighting has sent energy prices surging worldwide and stocks plummeting, and threatens the food supply and livelihoods of people around the globe who rely on crops farmed in the fertile Black Sea region.
The UN human rights office reported 406 confirmed civilian deaths, but said the real number is much higher. The invasion has also sent 1.7 million people fleeing Ukraine.
Mr Zelensky has called for more punitive measures against Russia, including a global boycott of its oil exports, which are key to its economy.
“If (Russia) doesn’t want to abide by civilised rules, then they shouldn’t receive goods and services from civilisation,” he said in a video address.