The reported battles on Saturday came as the Russian defence ministry said fighters from its Wagner mercenary group had captured two more areas of Bakhmut.
Wagner has spearheaded Russia’s attempt to take the city – the main target of Russia’s offensive in eastern Ukraine – since last summer.
The fight has been the longest and deadliest battle of the war for both sides.
“Bloody battles unprecedented in recent decades are taking place in the middle of the city’s urban area,” said Serhiy Cherevatyi, spokesperson for Ukraine’s eastern military command.
“Our soldiers are doing everything in bloody and fierce battles to grind down [the enemy’s] combat capability and break its morale,” he told the 1+1 television channel. “Every day, in every corner of this city, they are successfully doing so.”
Russia meanwhile claimed advances in Bakhmut.
“Wagner assault units have successfully advanced, capturing two districts on the northern and southern outskirts of the city,” Russia’s defence ministry said in a briefing.
It added that Russian army paratroop units were supporting the claimed advance by holding back Ukrainian forces on the flanks.
Al Jazeera could not independently confirm the reports.
The United Kingdom said in an intelligence update on Friday that Ukrainian troops had been forced to cede some territory in Bakhmut as Russia mounted a renewed assault there, with intense artillery fire over the previous two days.
‘Indescribable grief’
In nearby Sloviansk, which lies about 45km (27 miles) to the northwest, Ukrainian officials said rescue teams had recovered the bodies of two more people from under the rubble of a house hit in a Russian missiles attack, raising the toll there to 11.
According to Ukrainian officials, seven Russian missiles hit five buildings, five homes and an administrative building in the attack on Friday.
A previous toll reported nine dead, including a two-year-old boy who was rescued from the rubble but died on his way to the hospital, and 21 wounded.
Ukraine’s First Lady Olena Zelenska sent her condolences to the child’s family during this “indescribable grief”.
President Volodymyr Zelenskyy denounced Russia for “brutally shelling” residential buildings and “killing people in broad daylight”.
Ukrainian officials said rescuers in Sloviansk were continuing the search for five people who remained in the wreckage of the apartment building, as well as the residents of three units who were reported missing.
Ukraine’s air force meanwhile said the country would soon have weapons with which to try to prevent attacks like the one on Friday.
The delivery of the Patriot air defence system promised by the United States was expected in Ukraine sometime after Easter, Ukrainian air force spokesperson Yuriy Ihnat said.
The primarily Orthodox Christian country is preparing to observe Easter on Sunday.
Speaking on Ukrainian state TV, Ihnat declined to give a precise timeline for the arrival of the defensive missile system but said the public would know “as soon as the first Russian aircraft is shot down”.
Germany and the Netherlands also have pledged to provide a Patriot system each to Ukraine.
In addition, a SAMP/T antimissile system pledged by France and Italy “should enter Ukraine in the near future”, according to Ihnat.
On the diplomatic front, Brazil’s President Luiz Inacio Lula da Silva said on Saturday that the US should stop “encouraging war” in Ukraine, as he wrapped a state visit to China.
He also urged the European Union to “start talking about peace”.
Zelenskyy has said he will not negotiate with Russia as long as President Vladimir Putin is in power, and on Saturday, he reiterated Ukraine’s desire to join NATO as soon as possible.
Kyiv would need effective security guarantees before that happened, he said, but gave no details.
Russia had said this month that it wants any peace talks on Ukraine to focus on creating a “new world order”.
Moscow has long said it was leading a struggle against Washington’s dominance over the international stage and argues the Ukraine offensive is part of that fight.