The Russian defence ministry said its forces had secured two blocks in western districts and airborne units were providing reinforcements to the north and south. Russia sees Bakhmut as a stepping stone to more advances in eastern Ukraine.
Ukrainian Colonel General Oleksandr Syrskyi shared images on Sunday on the Telegram messaging app of him poring over a map with three other uniformed men, with the caption: "Bakhmut frontline. Our defence continues.
"We hit the enemy, often unexpectedly for him, and continue to hold strategic lines," he wrote.
Reuters was unable to verify battlefield reports.
Yevgeny Prigozhin, head of the private Wagner military force which is leading the Bakhmut assault, has claimed 80 per cent control of the city. Kyiv has repeatedly denied claims its troops are poised to withdraw.
Ukrainian President Volodymyr Zelenskiy vowed to keep defending the city.
"It is impossible for us to give up on Bakhmut because this will expand the battle front and will give the Russian forces and Wagner chances to seize more of our lands," Zelenskiy said in an interview with Al Arabiya news channel published on Sunday.
Also on Sunday, the Russian-installed head of Ukraine's southern Kherson region denied a report by a US think tank that Ukrainian forces had taken up positions on the Dnipro river's eastern bank.
"There is no enemy foothold on the left (eastern) bank of the Dnipro river ... our military completely controls that territory," Vladimir Saldo wrote on his Telegram channel.
Citing Russian military bloggers embedded with Moscow's forces, the Institute for the Study of War said Ukraine had "established positions" on the eastern bank, though it was not clear "at what scale or with what intentions".
Russia withdrew forces from the western bank last year as part of a series of withdrawals that signified a shift in momentum in Kyiv's favour.
Natalia Humeniuk, a spokeswoman for Ukraine's southern command, neither confirmed nor denied the report, which she told Ukrainian television foreshadowed "very powerful shelling" in districts around the west bank cities of Kherson and Beryslav.
"Reacting to such information, the enemy has significantly intensified its attacks on the opposite bank," she said. Civilians had been injured and about 30 buildings destroyed, including a school, she added.Â
