Ukrainian servicemen walk in a trench at a position near a frontline in Donetsk Region, Ukraine. Reuters
"Our heroes do not give up positions in Severodonetsk. In the city, fierce street fighting continues," Zelensky said in his nightly video address on Monday.
"And the Ukrainian Donbas stands, stands strong," he added referring to the region where Severodonetsk is located.
It was unclear which side had the upper hand, with "the situation changing from hour to hour", Oleksandr Stryuk, head of administration in Sievierodonetsk, said on television.
Russia denies targeting civilians in the conflict.
The city has become the main target of the Russian offensive in the Donbas — comprising Luhansk and Donetsk provinces — as the Kremlin's invasion grinds on in a war of attrition that has seen cities laid waste by artillery barrages.
Ukraine's defence ministry said Russia was also throwing troops and equipment into its drive to capture the largest remaining Ukrainian-held city in Luhansk.
Luhansk Governor Serhiy Gaidai said earlier on Monday the situation had worsened after Ukrainian defenders had pushed back the Russians over the weekend as they seemed close to victory.
In its nightly update, the Ukrainian military said two civilians were killed in Russian shelling in the Donetsk and Luhansk regions on Monday and that Russian forces had fired at more than 20 communities.
A view shows a residential building destroyed by a Russian military strike in Bakhmut, Donetsk. Reuters
Reuters could not independently verify the battlefield reports. Russia denies targeting civilians in the conflict.
Russia says it is on a mission to "liberate" the Donbas — partly held by separatist proxies of Moscow since 2014 — after Ukrainian forces pushed its troops back from the capital Kyiv and Ukraine's second city Kharkiv in the war's early stages.
Russia invaded Ukraine on Feb. 24, but calls its action a "special military operation" to stamp out what it sees as threats to its security. Ukraine and its Western allies call this a baseless pretext for a war to grab territory that risks turning into a wider European conflict.