The capture of the town in the north of Donetsk region could pave the way for Ukraine to make inroads into the adjacent Luhansk province, foiling Russia's goal of seizing all of the industrial Donbas region declared after its forces failed to subdue the entire country in February, military analysts said.
The regions are among four chunks of eastern and southern Ukrainian territory that Putin is expected on Friday to declare Russian-annexed land after what Ukraine and its allies say were bogus referendums staged at gunpoint.
Putin has said Russia could use nuclear weapons to defend Russian territory if necessary.
Ukraine has said it will not be swayed by such threats and will press ahead with its plans to drive all Russian forces out of Ukraine.
Lyman has served for months as a logistics and transport hub anchoring Russian operations in Donetsk region's north and its capture would be Ukraine's biggest gain since a lightning counter-offensive retook swathes of Kharkiv region this month.
The window is narrowing for Ukraine to make major advances before the northern hemisphere winter sets in, slowing down operations and giving Russia, which has declared a partial mobilisation, more time to fortify its lines.
Konrad Muzyka, director of the Rochan military consultancy in Poland, said Russian forces were trying to hold on in Lyman to buy time to prepare defensive lines and that the town would fall to Ukraine.
"The Russians are trying to delay the Ukrainian actions as much as possible so that they can create or upgrade their line of defence between Sievierodonetsk and the border with Russia," he said.
President Volodymyr Zelenskiy told Ukrainians in his nightly speech on Tuesday that there was "good news" from the front although he did not elaborate.
Luhansk region's exiled governor said this week: "When the situation around Lyman is resolved, then we can closely watch the actions of the Ukrainian army because Lyman is close to the river and on the other bank begins Luhansk region."
"The weather doesn't really play into our hands but in general, I still think we'll get the result soon," he said.
On Thursday, a Russian proxy official in Donetsk region said it was "pretty tense" and "difficult" in Lyman and that Ukrainian forces were constantly trying to attack.
"Of course we understand that such attempts will continue, at the moment our units are able to repel all these attacks," the official was quoted as saying by Russia's Tass news agency.
Lyman has seen its role as a hub for Russian forces diminish since Russian troops were routed in northeast Kharkiv region, analysts said.Â
The town had a population of 20,000 before Russia invaded on February 24.
"Ukrainian forces have made substantial gains around Lyman in the past 24 to 48 hours, with both Russian and Ukrainian sources indicating that Ukrainians have advanced west, north and even northeast of Lyman," said Karolina Hird, a researcher at the Institute for the Study of War.
"The envelopment of these positions and the collapse of this pocket around Lyman may allow - depending on how Ukrainian forces decide to pursue further gains - to unhinge this line and open up potential further advances to the east," she said.
Muzyka said it was still not yet clear how Russia's mobilisation of hundreds of thousands of reservists to plug big gaps in manpower would play out.
"If Russia starts deploying this personnel en masse, then it is going to have at least a short-term effect on the battlefield. It may decrease the tempo of Ukrainian attacks," he said.
