Russia's Defence Ministry said on Thursday its forces had taken four blocks in northwestern, western and south-western Bakhmut, Russian state-owned news agency RIA reported.
Reuters could not independently confirm the claim.
Russian forces have been struggling for months to capture the city in eastern Ukraine.
Hours before the Defence Ministry's update, the head of the Wagner mercenary group said it was suspending artillery fire in Bakhmut to allow Ukrainian forces on the other side of the front line to safely show the city to visiting US journalists.
Reuters could not independently verify Wagner's plans.
Wagner has been spearheading Russia's assault on Bakhmut since last summer in the longest and bloodiest battle of the war.
But Ukrainian forces have so far thwarted its attempts to take full control of the city.
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