Lysychansk is the new focus of the war after its twin city of Sievierodonetsk fell on Saturday, in a victory for Moscow's campaign to seize the eastern provinces of Luhansk and Donetsk on behalf of pro-Russian separatists.
Tass news agency on Sunday quoted a separatist official as saying Moscow's forces had entered Lysychansk on Monday from five directions and were isolating Ukrainian defenders. Reuters could not confirm the report.
The general staff of Ukraine's Armed Forces said Russian forces were using artillery to try to cut off Lysychansk from the south but made no mention of separatists entering the city.
Elena, an elderly woman from Lysychansk, was among dozens of evacuees who arrived in the Ukrainian-held town of Pokrovsk by bus from frontline areas.
"Lysychansk, it was a horror, the last week. Yesterday we could not take it any more," she said. "I already told my husband if I die, please bury me behind the house."
The RIA agency quoted a separatist official as saying separatist forces had evacuated more than 250 people, including children, on Sunday from Sievierodonetsk's Azot chemical plant.
The industrial area was the last part of Sievierodonetsk held by Ukrainian forces before they withdrew on Saturday after weeks of heavy fighting which left the town in ruins.
In Kyiv, Russian missiles struck an apartment block and close to a kindergarten on Sunday as world leaders gathered in Germany to discuss further sanctions against Moscow.
Deputy mayor Mykola Povoroznyk said one person was killed and six wounded in the first Russian attack on the capital in weeks.
In his nightly address, Ukrainian President Volodymyr Zelenskiy said a wounded seven-year-old girl was pulled from the rubble of a nine-storey apartment block. The girl's father was killed in the strike, he said.
"She was not threatened by anything in our country. She was completely safe, until Russia itself decided that everything was equally hostile to them now - women, children, kindergartens, houses, hospitals, railways," Zelenskiy said.
A Ukrainian air force spokesperson said the strike was carried out with four to six long-range missiles fired from Russian bombers more than 1000km away in southern Russia.
US President Joe Biden called the strikes acts of "barbarism", as leaders from the Group of Seven nations gathered for a summit in Germany. Russia denies targeting civilians.
Britain, Canada, Japan and the US have proposed a ban on imports of gold from Russia, aimed at wealthy Russians who have been buying safe-haven bullion to reduce the financial impact of Western sanctions.
The G7 leaders are also expected to discuss a possible price cap on Russian oil and efforts to tackle soaring global food and energy prices.
On Sunday, Russian missiles struck the central city of Cherkasy, which until now had been largely untouched by bombardment, according to authorities who said one person was killed and five others wounded.
Ukrainian presidential adviser Oleksiy Arestovych said the attack also hit a strategic bridge linking western Ukraine and the eastern battlefields.
"They are trying to limit the transfer of our reserves and Western weapons to the east," he told Reuters.
Russia's defence ministry said it had used high-precision weapons to strike military training centres in the regions of Chernihiv, Zhytomyr and Lviv, an apparent reference to strikes reported by Ukraine on Saturday. There was no immediate comment about Sunday's strikes on Kyiv or Cherkasy.
