The Wagner Group said it had control of the salt-mining town late on Tuesday night, but uncertainty surrounded the claim on Wednesday morning as battles continued to rage in the centre.
Ukrainian military officials said Soledar was one of several places in the Donetsk region being shelled and reported that their soldiers were still resisting the attack despite heavy fighting.
Head of the mercenary Wagner group Yevgeny Prigozhin said in a statement: “Wagner units took control of the entire territory of Soledar.
“A cauldron has been formed in the centre of the city in which urban fighting is going on.”
He stressed that only the private mercenary fighters - who are not part of the official Russian armed forces - were taking part “in the storming” of the town.
But the spokesman for the Eastern Group of the Armed Forces of Ukraine Serhiy Cherevaty countered the claim.
He said: “Russians say that [the town] is under their control, it is not true.”
Russia capturing Soledar, which is home to huge salt mines, would have huge military and commercial value for the Kremlin.
The UK Ministry of Defenceon earlier said Russian troops and Wagner fighters had probably taken control of most of the area following four days of advances.
The pair had been helping with civilian evacuation and humanitarian efforts in the war-torn Donetsk region.
It comes as Volodymyr Zelensky appeared in a video message during the 80th Golden Globes and thanked his allies in the “free world”.
The Ukrainian President was introduced by actor Sean Penn.
He addressed the crowd by saying: “The war in Ukraine is not over yet, but the tide is turning, and it is already clear who will win.
“There are still battles and tears ahead, but now I can tell you who was the best in the previous year – you in the free world, who united around support of free Ukrainian people.”
He added: “There will be no third world war, it’s not a trilogy.”
Russian Human Rights Commissioner Tatiana Moskalkova today confirmed an exchange of 40 prisoners of war had been agreed with Ukraine.
Ms Moskalkova met her Ukrainian counterpart Dmytro Lubinets at a hotel in Ankara on the sidelines of an international ombudsman conference to agree the swap.