An agreement has been reached for the establishment of 10 humanitarian corridors to evacuate civilians from front-line hotspots in Ukrainian towns and cities, officials said.
Deputy Prime Minister Iryna Vereshchuk said Ukraine had made the deal with Russia for the corridors to open immediately.
Russia has yet to comment or publicly verify the agreement.
Speaking on national television, Ms Vereshchuk said civilians trying to flee the city of Mariupol would have to leave in private cars as Russian forces were not letting buses through their checkpoints.
Previous attempts to arrange safe passage out of the southern port city, which is surrounded by Russian forces, have failed.
Ms Vereshchuk said that more than 100,000 people still needed to be evacuated from Mariupol.
Also on Saturday, Mariupol's Mayor said he had spoken to France's ambassador to Ukraine about options for evacuating civilians after French President Emmanuel Macron said he would propose to Russia a plan to help people leave.
Speaking on national television, Vadym Boichenko said the situation in the encircled city remained critical, with street fighting taking place in its centre.
Mariupol, which is normally home to about 400,000 people, has been under heavy bombardment for weeks.
Civilians trapped there have been sheltering in basements with little food, power or running water.
Meanwhile, Kyiv Mayor Vitali Klitschko announced that the curfew in the Ukrainian capital would be extended from 8pm (local time) on Saturday until 8am on Monday.
Mr Klitschko said the decision was made by the Ukrainian military, without giving further details.
Russians seize Slavutych
The governor of the Kyiv region said Russian forces had seized Slavutych, a town that sits just outside the exclusion zone that was established around the Chernobyl nuclear power plant after the 1986 disaster.
Oleksandr Pavlyuk said the Russians had also kidnapped the city's mayor, but media reported later in the day that he was released swiftly.
Neither claim could be verified independently.
Mr Pavlyuk said residents took to the streets with Ukrainian flags to protest the Russian invasion.
Reuters could not independently verify the reports.
Reports of rocket strikes in Lviv
The mayor of Lviv said another rocket had hit the city in western Ukraine on Saturday, not long after two rockets struck its outskirts in what appeared to be the first attacks within the city's limits since the start of the war with Russia.
Lviv, some 60 kilometres from the Polish border, has so far escaped the bombardment and fighting that has devastated some Ukrainian cities closer to Russia since Moscow launched its invasion on February 24.
But on Saturday, Governor Maksym Kozytskyy said two rockets had struck the city's eastern outskirts in the mid-afternoon and ordered residents to take shelter.
Later, Mayor Andriy Sadoviy said there had been another strike.
"One more rocket strike on Lviv," he said in an online post.
He did not share details of the location. He said the strike had damaged infrastructure but not residential buildings.
The first strikes set fire to an industrial facility storing fuel, but had not hit residential areas, Mr Sadoviy said earlier.
Governor Kozytskyy said five people had been wounded in that attack, citing preliminary figures.
Reuters witnesses in central Lviv saw heavy black smoke rising from the north-east side of the city and a strong smell of burning filled the air.
Men huddled together on the street to watch a plume of smoke rising behind an apartment block.
Most residents appeared to stay indoors, peeking out from behind curtains as others hurried past on the road carrying their bags.
ABC/wires
