The retreat of Ukrainian troops from Lysychansk and Sievierodonetsk, the last two Ukrainian-held cities in Luhansk, would bring Russia closer to one of its key war aims of capturing all of that region.
Divided by a river, the cities have become a key battleground in Russia's assault on the industrial heartland of Donbas and the fighting is nearing a "fierce climax," a top official said on Wednesday.
The General Staff of Ukraine's armed forces on Thursday confirmed the loss of Rai-Oleksandrivka and Loskutivka, about 5km from Lysychansk, and said that Russian troops were trying to surround Ukrainian forces there.
It did not say how it would respond but Governor Gaidai said Ukrainian troops might have to pull back.
"In order to avoid encirclement, our command could order that the troops retreat to new positions," he said on television.
"All of Lysychansk is within reach of their fire. It is very dangerous in the city."
The Ukrainian military almost never shares details concerning its strategy but chief commander of the armed forces Valery Zaluzhny conceded that Ukraine was having to make defensive adjustments.
"We are forced to conduct a mobile defence, to occupy more advantageous lines and positions," Zaluzhny said in an online post that did not name specific areas.
"The price of freedom is high."
In an evening update, the Ukrainian military said Russian forces backed up with additional reserves had resumed their offensive south of Lysychansk, after Ukrainian troops earlier pushed them back.
The war has entered a brutal attritional phase in recent weeks, with Russian forces concentrating overwhelming artillery firepower on this pocket of the Donbas, which it claims on behalf of separatists.
Gaidai said Lysychansk could still be reached by road, allowing civilians to leave the city.
Russia's TASS news agency had earlier cited Russian-backed separatists as saying the city was surrounded and cut off from supplies.
Meanwhile a senior Ukrainian defence official said Ukrainian troops were holding out in a part of Sievierodonetsk, which is mostly controlled by Russian forces.
Despite damage to supply routes, these troops could still receive weapons and ammunition and move out the wounded, Oleksiy Gromov, Deputy Chief of the Main Operational Department of the General Staff, said.
The UK defence ministry said on Thursday Russia's recent advances were putting the Ukrainian forces holding out in Lysychansk-Sieverodonetsk under increasing pressure.
"However, its (Russia's) efforts to achieve a deeper encirclement to take western Donetsk Oblast (region) remain stalled," the ministry said in a daily update on Twitter.
