A withdrawal could bring Russian President Vladimir Putin closer to his goal of capturing eastern Ukraine's Luhansk and Donetsk regions in full.
His troops have gained ground in the two areas collectively known as the Donbas while blasting some towns to wastelands.
Luhansk's governor Serhiy Gaidai said Russian troops had entered Sievierodonetsk, the largest Donbas city still held by Ukraine, after trying to trap Ukrainian forces there for days.
However he added Russian forces would not be able to capture the Luhansk region "as analysts have predicted".
"We will have enough strength and resources to defend ourselves. However, it is possible that in order not to be surrounded we will have to retreat," Gaidai said on Telegram.
Gaidai said 90 per cent of buildings in Sievierodonetsk were damaged with 14 high-rises destroyed in the latest shelling.
Russia's separatist proxies said they controlled Lyman, a railway hub west of Sievierodonetsk. Ukraine said Russia had captured most of Lyman but that its forces were blocking an advance to Sloviansk, to the southwest.
President Volodymyr Zelenskiy said Ukraine was protecting its land "as much as our current defence resources allow".
Ukraine's military said it had repelled eight attacks in Donetsk and Luhansk on Friday, destroying tanks and armoured vehicles.
"If the occupiers think that Lyman and Sievierodonetsk will be theirs, they are wrong. Donbas will be Ukrainian," Zelenskiy said in an address.
The General Staff of Ukraine's armed forces said on Saturday Ukrainian forces had repelled eight assaults in the Donetsk and Luhansk regions in the previous 24 hours.
Russia's attacks included artillery assaults in the Sievierodonetsk area "with no success", it said.
Analysts at the Institute for the Study of War, a Washington-based think tank, said while Russian forces had begun direct assaults on built-up areas of Sievierodonetsk, they would likely struggle to take ground in the city itself.
"Russian forces have performed poorly in operations in built-up urban terrain throughout the war," they said.
Russian troops advanced after piercing Ukrainian lines last week in the city of Popasna, south of Sievierodonetsk.
Russian ground forces have captured several villages northwest of Popasna, Britain's defence ministry said.
Reached by Reuters journalists in Russian-held territory on Thursday, Popasna was in ruins. The bloated body of a dead man in combat uniform could be seen lying in a courtyard.
Resident Natalia Kovalenko had left the cellar where she was sheltering in the wreckage of her flat, its windows and balcony blasted away. She said a shell hit the courtyard, killing two people and wounding eight.
"We are tired of being so scared," she said.
Russia's eastern gains follow the withdrawal of its forces from approaches to the capital, Kyiv, and a Ukrainian counter-offensive that pushed its forces back from Ukraine's second city, Kharkiv.
Russian forces shelled parts of Kharkiv on Thursday for the first time in days, killing nine people, authorities said. The Kremlin denies targeting civilians in what it calls its "special military operation".
Ukraine's General Staff said on Saturday while there was no new attack on the city, there were multiple Russian strikes on nearby communities and infrastructure.
In the south, where Moscow has seized a swath of territory since the February 24 invasion, including the port of Mariupol, Ukrainian officials say Russia aims to impose permanent rule.
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