Russian forces intensified their assault on Severodonetsk on Sunday as they edge closer to seizing all of the Luhansk region.
The Kremlin has launched an all-out assault to encircle the twin cities of Severodonetsk and Lysychansk in recent days.
Experts warned Russia’s capture of the region could see Moscow control large parts of the east of the country.
The Russian defence ministry said its troops were in full control of Lyman, the site of a railway junction west of the Siverskyi Donets River in the Donetsk.
Sievierodonetsk is 40 miles northeast of Lyman on the eastern side of the river.
It is the largest city in the Donbas held by Ukraine and is under heavy assault.
"The situation has extremely escalated,” said Serhiy Gaidai, the governor of Luhansk.
The shelling was so intense it was not possible to assess casualties and damage, though two people were killed on Saturday and 13 more buildings in the city were destroyed, he said.
Gaidai said on Friday that Ukrainian troops might have to retreat from the city to avoid capture but it was not clear whether they had begun to pull out.
Russian artillery was also pounding the Lysychansk-Bakhmut road, which Russia must take to close a pincer movement and encircle Ukrainian forces, and police said there was "significant destruction" in Lysychansk.
President Zelensky said the military situation in the Donbas was very complicated.
He added defences were holding up in a number of places, including Sievierodonetsk and Lysychansk.
"It’s indescribably difficult there. And I am grateful to all those who withstood this onslaught," he said.