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Ukraine is set to pull its troops from the ruined city of Sievierodonetsk after weeks of street fighting and bombardment, the regional governor says, in what would be a significant gain for Russia as it grinds out its offensive in the east.

Russian troops also occupied a town about 10km further south, both sides said on Friday, as Russia closed in on the last slivers of Ukrainian-held territory in the industrial region of Luhansk.

Russia said it had encircled about 2000 Ukrainian and what it called foreign troops in the area.

Reuters could not independently verify any of the battlefield accounts.

The reports came four months to the day since Russian President Vladimir Putin sent tens of thousands of troops over the border, unleashing a conflict that has killed thousands, uprooted millions and reduced whole cities to rubble.

If it goes ahead, the withdrawal from Sievierodonetsk would mark the biggest reversal for Ukraine since the loss of the southern port of Mariupol in May.

The latest Russian advances appeared to bring the Kremlin closer to taking full control of Luhansk, one of Russia's stated war objectives, and set the stage for Sievierodonetsk's twin city of Lysychansk to become the next main focus of fighting.

Luhansk regional governor Serhiy Gaidai said troops in Sievierodonetsk had already received the order to move to new positions.

"Remaining in positions smashed to pieces over many months just for the sake of staying there does not make sense," Gaidai said on Ukrainian television.

Russia invaded Ukraine on February 24 but abandoned an early advance on the capital Kyiv in the face of fierce resistance bolstered by foreign arms.

Since then Russia and its proxies have focused on the south and Donbas, an eastern territory made up of Luhansk and its neighbour Donetsk, deploying overwhelming artillery in some of the heaviest ground fighting in Europe since World War II.

Ukrainian forces had held out for weeks in Sievierodonetsk, trying to wear down Russian troops through attrition and buy time for the arrival of heavy weapons supplies.

"Our forces had to withdraw and conduct a tactical retreat because there was essentially nothing left there to defend. There was no city left there and, secondly, we could not allow them to be encircled," Oleksander Musiyenko, a Kyiv-based military analyst, said.

Ukraine on Friday again pressed for more arms, with its top general, Valeriy Zaluzhniy, telling his US counterpart in a phone call that they needed "fire parity" with Russia to stabilise the situation in Luhansk.

Ukraine's Defence Ministry said the Russian forces were trying to surround Lysychansk and mounting assaults on Sievierodonetsk to win full control.

But spokesman Oleksandr Motuzyanyk declined to comment on Gaidai's remarks about a withdrawal.

Further south, Russian troops had entered the town of Hirske and fully occupied the surrounding district on Friday, municipal head Oleksiy Babchenko said.

"There is a red flag flying over the municipal administration (in Hirske)," a spokesperson for the regional administration told Reuters by telephone.

Russia's defence ministry said it had taken Hirske and the nearby town of Zolote after what it described as a "rout" of Ukrainian soldiers.

It earlier said it had encircled up to 2000 Ukrainian troops, including 80 foreign fighters, at Hirske.

Vitaly Kiselev, an official in the Interior Ministry of the separatist Luhansk People's Republic - recognised only by Russia - told Russia's TASS news agency that there were about 4500 Ukrainian servicemen in the area taken over by Russian and separatist forces in Hirske but did not say what had happened to them.

Kiselev said it would take another week and a half to secure full control of Lysychansk and that an unknown number of people remained holed up and did not want to leave the Azot chemical plant in Sievierodonetsk.

The general staff of Ukraine's armed forces said its troops had some success in the southern Kherson region, forcing the Russians back from defensive positions near the village of Olhine, the latest of several Ukrainian counter-assaults.

Ukraine's foreign minister played down the significance of the possible loss of more territory in the Donbas.

"Putin wanted to occupy the Donbas by May 9. We are (there) on June 24 and still fighting. Retreating from a few battles does not mean losing the war at all," Dmytro Kuleba said in an interview with the Italian newspaper Corriere della Sera.

Russia says it sent troops into Ukraine to degrade its southern neighbour's military capabilities and root out people it called dangerous nationalists.

Ukraine says Russia has launched an imperial-style land grab.

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