"At the moment, Lyman is partially surrounded," said Denis Pushilin, the Russian-installed governor in the Donetsk region.
"The Svatove road is under our control, but under fire periodically," he wrote on Telegram, adding that the nearby villages of Yampil and Drobycheve "are not under total control".
In his nightly address on Thursday, Zelenskyy had announced there was "good news" from the front, without elaborating.
Lyman is located around 160 kilometres southeast of Kharkiv, Ukraine’s second-largest city. For months, it has served as a logistics and transport hub anchoring Russian operations in the Donetsk region.
The Institute for the Study of War says that losing the city would be a major blow to Moscow's war effort.
“The collapse of the Lyman pocket will likely be highly consequential to the Russian grouping [and] may allow Ukrainian troops to threaten Russian positions along the western Luhansk,” the Washington-based institute said.
If Kyiv recaptures Lyman, it would be their biggest successful counteroffensive since taking swathes of the Kharkiv region earlier this month.
Putin has made it a priority for Russian forces to control all of the industrial Donbas regions and his "partial mobilisation" of Russian troops aims to fortify Russian positions.
In addition to Lyman, Russian occupation forces in the neighbouring Kherson region reported on Friday that a senior government official was killed in a Ukrainian strike.
Alexei Katerinitshev, a deputy head of the Moscow-backed administration, was killed "in a precise strike" on his home on Thursday night, according to the Russian news agency TASS.
Ukrainian and Russian forces have blamed each other for the attack, which occurred near a crossing point between
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