Approximately 130 survivors were reportedly found alive and rescued on Thursday from a theater bombed by Russia in Ukraine's besieged city of Mariupol.

More than 1,000 civilians had been sheltering in the Mariupol theater struck on Wednesday, according to Mayor Vadim Boychenko.

"People are coming out alive!" Ukrainian lawmaker Serhiy Taruta wrote on Facebook on Thursday, after rubble began being cleared.

Kyiv and Moscow have blamed each other for the incident, with the Ukrainian side saying it was a deliberate Russian bombing, and Russia accusing the Ukrainian nationalist Azov regiment of the attack.

Conditions are dire in Mariupol, which is blocked on all sides by Russian forces, preventing food, water or medicines from reaching the local population.

Private cars are now being allowed out of the port city, Boychenko said earlier on Thursday, after some 6,500 cars had left in the last two days while the city was under fire.

According to Kyiv, more than 6,000 residents of Mariupol had left the city in 24 hours, including 2,000 children.

In the northern city of Chernihiv, meanwhile, local authorities reported that dozens more had died in Russian attacks.

"In the last 24 hours alone, 53 bodies of our citizens murdered by the Russian aggressor have arrived at the city's morgues," the head of the military administration, Vyacheslav Chaus, wrote on Telegram.

He said that further negotiations were planned on a possible escape corridor.

In the capital, at least one person was killed and three injured when debris from an intercepted missile hit a high-rise building in the east of the city overnight, local authorities said.

The United Nations High Commissioner for Human Rights has documented the deaths of at least 780 civilians in Ukraine since Russian troops invaded three weeks ago, it announced, with 58 children and young people among the victims.

The office had also verified information about 1,252 wounded people. Actual casualty numbers are thought to be far higher.

Across Ukraine, an overall nine humanitarian corridors were intended to be set up on Thursday, Deputy Prime Minister Iryna Vereshchuk said, but previous planned evacuation attempts have repeatedly failed.

Negotiations between Kyiv and Moscow were continuing online, according to Kremlin spokesperson Dmitry Peskov.

"Our delegation is making great efforts and showing far more willingness than our Ukrainian counterparts," he added, arguing that Russia's conditions were "extremely clear."

He rejected media reports on the possibility of direct talks between Ukrainian President Volodymyr Zelenskyy and Russian President Vladimir Putin, arguing that these were "not correct."

The Financial Times daily reported Wednesday that a 15-point plan was being drafted, topped with Moscow's demand for Ukrainian neutrality and demilitarization, as well as Kyiv's insistence that Russian troops withdraw.

The Kremlin also hit back at U.S. President Joe Biden's claim that Putin is a "war criminal," with Peskov calling the assertion "unacceptable and unforgivable."

Russia further rejected an order from the International Court of Justice to halt the use of military force in Ukraine.

On Friday, Biden is due to hold a phone call with Chinese leader Xi Jinping to discuss "Russia’s war against Ukraine" and other issues, the White House said in a statement.

Relentless Russian shelling hit houses in the northeastern Ukrainian region of Kharkiv overnight, the government in Kyiv said, adding that there was an unknown number of casualties.

The Russian "occupiers" had also destroyed a school and a cultural center in the town of Merefa, it said.

The Ukrainian army reported heavy explosions in the southern Kherson region, which has been under Russian control since the early days of the war.

The army warned that electricity, water and gas supplies were facing increasing disruptions across much of the country and that shortages of food and medicine were a growing concern.

In Luhansk meanwhile, separatist leaders said the Ukrainian army had shelled four settlements in the self-declared republic, Russian news agency Tass reported. The information could not be independently verified.