Ukraine has ordered some residents to leave Kupiansk, as Russia seeks to re-take the city it left last year.
Kharkiv’s regional authorities said families with children and people “with limited mobility” must leave due to “constant” shelling by Russian forces.
Russia seized the north-eastern city early in the full-scale invasion, with Ukraine recapturing it last September.
Meanwhile, Russian mercenaries say they have “practically encircled” the key city of Bakhmut in eastern Ukraine.
The comments were made by Yevgeniy Prigozhin, who heads the paramilitary Wagner group.
In its latest news bulletin, Ukraine’s military said Russian troops continued their offensive on Bakhmut – but the attacks were “repelled”.
Earlier this week, Ukrainian President Volodymyr Zelensky admitted the situation in Bakhmut – about 130km (80 miles) south-east of Kupiansk – was becoming “more and more difficult”.
In Kupiansk, the Kharkiv regional military administration said on Thursday the evacuation order was due to the “unstable security situation” caused by Russian shelling.
It said those evacuated would be provided with assistance, including accommodation, food, humanitarian aid and medical support.
Other citizens were also permitted to leave the region, it added. The city had a pre-war population of around 25,000.
The military said 812 children are currently registered in Kupiansk and the surrounding district, as well as 724 disabled people.
The Institute for the Study of War said this week that Russian forces were continuing “limited ground attacks” north-east of Kupiansk, as well as offensive operations around Kreminna, about 80km south-east.
Kupiansk – home to an important railway junction – has witnessed fierce fighting since the war began, with Russia taking control in a matter of days, and occupying the city for several months.
However, in September Ukrainian forces took back control amid a rapid counter-attack in the country’s north-east that saw almost entire of the Kharkiv region brought under Kyiv’s control.
Those advances – and the liberation of the southern city of Kherson – were the most significant front-line changes since Russia withdrew from areas around Kyiv in April.
Last month, Ukraine warned that Russia was preparing a major new offensive, with officials saying Moscow has amassed thousands of troops on Ukraine’s eastern flank.
And Defence Minister Oleksii Reznikov warned a month ago that Russia could “try something” to mark the anniversary of the full-scale invasion on 24 February.
But that apparent escalation has not translated into major successes on the battlefield, despite apparent advances around Kupiansk and Bakhmut.
In a video posted on Telegram, Mr Prigozhin, head of the paramilitary group Wagner, on Friday said that “pincers are closing in” around Bakhmut.
And in a direct message to President Zelensky, he said Wagner units had “practically encircled” the city, with only one road remaining. He called on the Ukrainian president 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 sabotage group of entering a Russian border region and opening fire on civilians.
The governor of Bryansk region said “saboteurs from Ukraine” had fired at a civilian car in Lyubechane, a border village, killing two men and wounding a 10-year-old boy.
Kyiv strongly denied the Russian claim, describing it as Moscow’s provocation.
