Both Ukraine and Russia have reported inflicting significant losses in the eastern city.
Ukraine and Russia have reported inflicting heavy losses as the battle for Bakhmut rages on.
Moscow has been trying to take the eastern Ukrainian city for months in a grinding war of attrition.
Ukrainian President Volodymyr Zelensky said Russian forces had suffered more than 1,100 deaths in the past few days, with many more seriously injured.
Russia said it had killed more than 220 Ukrainian service members over the past 24 hours.
The BBC is unable to verify the numbers given by either side.
Analysts say Bakhmut has little strategic value, but has become a focal point for Russian commanders who have struggled to deliver any positive news to the Kremlin.
Capture of the city would bring Russia slightly closer to its goal of controlling the whole of Donetsk region, one of four regions in eastern and southern Ukraine annexed by Russia last September following referendums widely condemned outside Russia as a sham.
“In less than a week, starting from 6 March, we managed to kill more than 1,100 enemy soldiers in the Bakhmut sector alone, Russia’s irreversible loss, right there, near Bakhmut,” Mr Zelensky said in his nightly video address.
He added that 1,500 Russian soldiers were wounded badly enough to keep them out of further action.
Russia’s defence ministry said Russian forces had killed “more than 220 Ukrainian servicemen”.
The commander of Ukraine’s ground forces, Col Gen Oleksandr Syrskyi, said the Russian mercenary Wagner Group was attacking his troops from several directions in a bid to break through defences and advance to the central districts of the town.
The paramilitary organisation is at the heart of the Russian assault on Bakhmut. Its leader, Yevgeny Prigozhin, has staked his reputation, and that of his private army, on seizing Bakhmut.
He said on Sunday that the situation in the city was “difficult, very difficult, the enemy is fighting for every metre”.
“And the closer to the city centre, the fiercer the fighting,” he said in a voice recording published on Telegram.
After his envisioned capture of Bakhmut, “we will begin to reboot” and “will start recruiting new people from the regions”, he said.
And on Saturday, the Institute for the Study of War – a US think tank – reported that Moscow’s offence was stalling.
“Wagner Group fighters are likely becoming increasingly pinned in urban areas… and are therefore finding it difficult to make significant advances,” it said.
There were about 70,000 people living in Bakhmut before the invasion, but only a few thousand remain. The city was once best known for its salt and gypsum mines and huge winery.
Like Russia, Ukraine has also given Bakhmut political significance, with President Zelensky making the city an emblem of resistance.
When he visited Washington in December, he called it “the fortress of our morale” and gave a Bakhmut flag to the US Congress.
Western officials estimate between 20,000 and 30,000 Russian troops have been killed or injured so far in and around Bakhmut.