At least seven civilians were reported killed in the Lugansk region where Severodonetsk is located and in the southern city of Mykolaiv, while a revered wooden church was reported to be on fire because of the fighting.
Lugansk regional governor Sergiy Gaiday said in an interview posted online that the invading forces had captured most of Severodonetsk, but that the Ukrainian military was pushing them back.
"The Russian army, as we understand, is throwing all its power, all its reserves in this direction," said Gaiday, who on Friday claimed Ukrainian troops had managed to win back a fifth of the city.
Russia's army however claimed some Ukrainian military units were withdrawing from the city.
The press service of Ukraine's presidential office said that "street fighting" was continuing in Severodonetsk and "assault operations are underway" in an industrial part of the city.
Severodonetsk is the largest city still in Ukrainian hands in the Lugansk region, where Russian forces have been gradually advancing in recent weeks after retreating or being repelled from other areas, including around the capital Kyiv.
Thousands of people have been killed, millions forced to flee and towns turned into rubble since Russian President Vladimir Putin ordered an all-out assault on his pro-Western neighbour on February 24.
Western powers have slapped increasingly stringent sanctions on Russia and supplied arms to Ukraine but divisions have emerged on how to react.
French President Emmanuel Macron on Friday said Putin had committed a "fundamental error" but said Russia should not be "humiliated" so that a diplomatic solution could be found.
Ukrainian Foreign Minister Dmytro Kuleba reacted on Saturday saying such calls "only humiliate France" and any country taking a similar position.
"It is Russia that humiliates itself. We all better focus on how to put Russia in its place. This will bring peace and save lives," he said.
Foreign volunteers killed
Regardless of diplomatic efforts, the conflict has raged in the south and east of the country.
Ukrainian officials on Saturday announced the death of four foreign military volunteers fighting Russian forces but did not specify when or under what circumstances they died.
The International Legion of Defence of Ukraine, an official volunteer brigade, named the men and published photos of them, saying they were from Germany, the Netherlands, Australia and France.
The deaths of the two men named from the Netherlands and Australia had already been reported and France's foreign ministry on Friday said a French volunteer fighter had been killed in combat.
Ukraine also reported two victims from a Russian missile strike on Odessa in the southwest, without specifying if they were dead or injured.
Russia's defence ministry said it had struck a "deployment point for foreign mercenaries" in the village of Dachne in the Odessa region.
It also claimed a missile strike in the northeastern Sumy region on an artillery training centre with "foreign instructors".
Apart from the human toll, the conflict has caused widespread damage to Ukraine's cultural heritage.
On Saturday, Ukrainian officials reported that a large Orthodox wooden church, a popular pilgrim site, was on fire and blamed Russian forces.
Russia continues to prove "its inability to be part of the civilized world," Culture Minister Oleksandr Tkachenko said in a statement.
Russia's defence ministry blamed "Ukrainian nationalists" for the blaze and said its forces were not operating in the area.
The church was built in 2009 on the site of another church that was blown up in 1947.
Grain exports 'no problem'?
Russian troops now occupy a fifth of Ukraine's territory and Moscow has imposed a blockade on its Black Sea ports.
The blockade has sparked fears of a global food crisis since Ukraine and Russia are among the top wheat exporters in the world.
The United Nations said it was leading intense negotiations with Russia to allow Ukraine's grain harvest to leave the country.
Putin in a televised interview Friday said there was "no problem" to export grain from Ukraine, via Kyiv- or Moscow-controlled ports or even through central Europe.
The UN has warned that African countries, which normally import more than half of their wheat consumption from Ukraine and Russia, face an "unprecedented" crisis.
Food prices in Africa have already exceeded those in the aftermath of the 2011 Arab Spring uprisings and the 2008 food riots.
On Friday, Putin met the head of the African Union, Senegalese President Macky Sall, at his Black Sea residence in Sochi.
After the meeting, Sall said he was "very reassured", adding that Putin was "committed and aware that the crisis and sanctions create serious problems for weak economies".