Ukraine has ordered some residents to leave Kopyansk, as Russia tries to retake the city, which it abandoned last year.
Kharkiv regional authorities said families with children and people with “restricted mobility” should leave due to “continuous” shelling by Russian forces.
Russia captured the northeastern city at the start of the full-scale offensive, only to be recaptured by Ukraine last September.
Meanwhile, Russian mercenaries say they have “virtually surrounded” the key city of Bakhmut in eastern Ukraine.
The comments were made by Yevgeniy Prigozhin, head of the paramilitary Wagner group.
In its latest news bulletin, the Ukrainian military said Russian troops continued their offensive on Bakhmut – but the attacks had been “repulsed”.
Earlier this week, Ukrainian President Volodymyr Zelensky admitted that the situation in Bakhmut – about 130 kilometers (80 miles) southeast of Kopyansk – was becoming “more and more difficult”.
In Kupiansk, Kharkiv’s regional military administration said Thursday that the evacuation was ordered because of the “volatile security situation” caused by Russian shelling.
The evacuees will be provided with assistance including shelter, food, humanitarian aid and medical assistance, he said.
He added that other citizens have also been allowed to leave the area. The pre-war population of the city was around 25,000.
The military said there are currently 812 children registered in Kapiansk and the surrounding district, as well as 724 people with disabilities.
The Institute for the Study of War said this week that Russian forces were continuing “limited ground offensives” northeast of Kopyansk, as well as offensives around Karimna, about 80 kilometers to the southeast.
Kopyansk – home to an important railway junction – has seen heavy fighting since the start of the war, with the Russians capturing the city within days and holding it for months.
However, in September Ukrainian forces retook control of the country’s northeast in a swift counteroffensive that saw almost the entire Kharkiv region fall under Kiev’s control.
Those developments – and the liberation of the southern city of Kherson – were the most significant front-line changes since Russia’s withdrawal from areas around Kiev in April.
After Ukraine warned last month that Russia was preparing a major new offensive, officials say Moscow has massed thousands of troops in eastern Ukraine.
And Defense Minister Oleksii Rezinkov warned a month ago that Russia could “try to do something” on the February 24 anniversary of the full-scale attack.
But despite apparent progress around Kopyansk and Bakhmut, this apparent increase did not translate into major battlefield successes.
In a video posted on Telegram, the head of the paramilitary group Wagner, Mr Prigozen, said on Friday that “the pincers are closing in” around Bachmut.
And in a direct message to President Zielinski, he said that Wagner units had “virtually surrounded” the city, with only one road remaining. They demanded the President of Ukraine to abandon the city.
In a separate development on Friday, Vladimir Putin is chairing a meeting with the country’s top security officials.
It comes a day after the Russian president accused a Ukrainian subversive group of entering the Russian border area and firing on civilians.
The governor of the Bryansk region said “saboteurs from Ukraine” had opened fire on a civilian’s car in the border village of Lyubichane, killing two people and wounding a 10-year-old boy.
Kiev vehemently denied this claim by Russia calling it a provocation by Moscow.
