"Luhansk has not been cut off," said the region's military administration head, Serhiy Haidai.
Fighting has reached the outskirts of the big city of Severodonetsk, a key Russian target.
If they capture the road to Bakhmut then the city could be surrounded.
Mr Haidai said that while the route to the south-west of Severodonetsk had come under fire, humanitarian aid was still getting through.
Six people were killed in Russian bombardment on Tuesday, he said on his Telegram feed.
If Russia captures the neighbouring town of Lysychansk then the whole Luhansk region will be in its hands.
The Luhansk regional head accused Russian forces of bombarding civilian infrastructure while 15,000 civilians remained in Severodonetsk, mainly in shelters.
"They can't take the city so they have decided to try to destroy it, and to make our troops leave the city," Mr Haidai said.
American volunteer combat medic Dallas Anthony Casey, who has helped evacuate civilians from Severodonetsk, said shelling of civilian areas was constant.
"Two days ago one of the volunteer ladies was hit when going door to door. She lost the lower half of her leg and lost a lot of blood before we could get to her. We had to do CPR, but we were unsuccessful," he told the BBC.
Three months after Russia launched its invasion, its war aims are currently focused on seizing Ukraine's eastern regions in the industrial heartland of Donbas.
Russia has sustained heavy losses on the battlefield and for the first time since the war began state TV showed President Vladimir Putin visiting casualties at a Moscow hospital. In the very brief clip filmed for a weekend TV program, the Russian leader could be heard telling a man "your father will be proud of you".
President Volodymyr Zelensky has described conditions in eastern Ukraine as extremely difficult while prominent Russian figures have indicated they are prepared for a protracted conflict. Security Council Secretary Nikolai Patrushev said Russia was not chasing deadlines and Defense Minister Sergei Shoigu has admitted the pace of the offensive is slowing down.
The Ukrainian leader appealed to European countries to show unity in the face of Russian aggression and Foreign Minister Dmytro Kuleba called on the West to "kill Russian exports" by stopping all trade.
As well as bombarding towns and cities in the east, Russia launched its biggest missile attack yet on the southern city of Zaporizhzhia at dawn on Wednesday, killing one person and damaging more than 60 homes and a shopping center.
In a clear statement of intent that Russia's long-term plans extend across the south, as well as eastern Ukraine, President Putin signed a decree making it easier for citizens in Russian-occupied areas of the south to get Russian passports.
While the region of Kherson is under Russian occupation, only parts of the neighbouring region of Zaporizhzhia are in Russian hands. A Russian-appointed official in occupied Zaporizhzhia, Vladimir Rogov, declared on Wednesday that the entire region should become a fully-fledged part of
Russia, rather than a so-called people's republic.
Occupied areas of Donetsk and Luhansk became proxy republics after they were captured in 2014 and Russia has already handed out 800,000passports there.
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