Russia also launched missile attacks on Saturday on areas far from the heart of the eastern battles.
Serhiy Haidai, governor of Luhansk province, said on Facebook on Saturday that Russian and separatist fighters tried to blockade Lysychansk from the south. The city lies next to Sievierodonetsk, which has endured a relentless assault and house-to-house fighting for weeks.
Russia's Interfax news agency quoted a spokesman for the separatist forces, Andrei Marochko, as saying Russian troops and separatist fighters had entered Lysychansk and that fighting was taking place in the heart of the city. There was no immediate comment on the claim from the Ukrainian side
Lysychansk and Sievierodonetsk have been the focal point of a Russian offensive aimed at capturing all of eastern Ukraine's Donbas region and destroying the Ukrainian military defending it, the most capable and battle-hardened segment of the country's armed forces.
The two cities and surrounding areas are the last major pockets of Ukrainian resistance in Luhansk, 95 per cent of which is under Russian and local separatist control. The Russians and separatists also control about half of Donetsk, the second province in the Donbas.
Russian bombardment has reduced most of Sievierodonetsk to rubble and cut its population from 100,000 to 10,000. Some Ukrainian troops are holed up in the huge Azot chemical factory on the city's edge, along with about 500 civilians. A separatist representative, Ivan Filiponenko, said forces evacuated 800 civilians from the plant during the night, Interfax reported.
After Haidai said Friday that Ukrainian forces had begun retreating from Sievierodonetsk, military analyst Oleg Zhdanov said some of the troops were heading for Lysychansk. But Russian moves to cut off Lysychansk will give those retreating troops little respite.
Some 1,000 kilometres to the west, four Russian rockets hit a "military object" in Yaroviv, Lviv regional governor Maksym Kozytskyy said. He did not give further details of the target, but Yaroviv has a sizeable military base used for training fighters, including foreigners who have volunteered to fight for Ukraine.
About 30 Russian rockets also were fired on the Zhytomyr region in central Ukraine on Saturday morning, killing one Ukrainian soldier, regional governor Vitaliy Buchenko said.
In the north, about 20 rockets were fired from Belarus into the Chernihiv region, the Ukrainian military said.
Ukraine's air command said Russian long-range Tu-22 bombers were deployed from Belarus for the first time. The neighbouring country hosts Russian military units and was used as a staging ground before Russia invaded Ukraine, but its own troops have not crossed the border.
A senior US defence official, speaking in Washington on condition of anonymity, on Friday called the Ukrainians' move out of Sievierodonetsk a "tactical retrograde" to consolidate forces into positions where they can better defend themselves. This will add to Ukraine's effort to keep Russian forces pinned down longer in a small area, the official said.
Following a botched attempt to capture Kyiv, Ukraine's capital, in the early stage of the invasion that started February 24, Russian forces have shifted their focus to the Donbas, where the Ukrainian forces have fought Moscow-backed separatists since 2014.