A fresh row has broken out between the Wagner Group and the Russian military, as long-simmering tension between the groups threatens to spill over.
Wagner boss Yevgeny Prigozhin dismissed Russia's claims to have inflicted heavy losses on Ukraine in the Donbas when Kyiv attempted offensive action.
Prigozhin said the claims were "simply wild and absurd science fiction".
But Russia's defence ministry doubled down on the claims on Tuesday, saying Ukraine suffered over 3,700 casualties.
Russia said the attack was part of a Ukrainian "long-promised offensive" in the Donetsk region, which took place over Sunday and Monday.
Ukraine's military said on Monday that it had no information about a major attack in the region, and that it wouldn't comment on claims it called "fake".
Speaking to state media, Russian Defence Minister Sergei Shoigu insisted that his forces had inflicted over 3,715 casualties on Ukraine during the attack and destroyed dozens of pieces of military equipment.
"The attempts at an offensive have failed. The enemy has been halted. Russian soldiers and officers displayed courage and heroism in the battles," Mr Shoigu said, adding that 71 Russian troops died in the fighting.
However, Prigozhin was quick to hit out at the defence ministry's claims on Monday evening, saying the claims of a huge success would amount to a "massacre".
"Therefore, I think that these are just some wild fantasies," he wrote on Telegram. He also accused Russian forces of "slowly" withdrawing from the village of Berkhivka on the outskirts of Bakhmut, calling it a "disgrace".
Moscow rejected Prigozhin's claims in a statement on Tuesday. It said his allegations did "not correspond to reality" and insisted that the suburb remained under Moscow's control.
Officials in Kyiv have admitted to shifting to "offensive actions" in some parts on the country and on Tuesday a US official told the BBC's American partner CBS News that it is accurate to say that the Ukraine counter offensive is in its opening phases, but that the main thrust has not yet begun.
On Monday, Kyiv's Deputy Defence Minister Hanna Maliar said some offensive operations had commenced around the eastern city of Bakhmut, which she called an "epicentre of hostilities".
In a video address late on Monday, President Volodymyr Zelensky thanked Ukrainian fighters for delivering "the news we are expecting" in the Bakhmut region.
Bakhmut has for months been at the heart of fierce fighting. It has little strategic value - but is important symbolically both for Kyiv and Moscow.
Prigozhin's fighters were heavily involved in the battle for Bakhmut, and he regularly criticised Russian officials for what he said was a lack of front-line support. But at the end of May he said his forces were withdrawing from the city in order to transfer control to the Russian army.
He is believed to hold his own political ambitions, and his forces' objectives have not always aligned with the military's, resulting in tension between the factions.
On Monday, the paramilitary group said it had detained a regular Russian army officer, who allegedly opened fire on a Wagner vehicle in Ukraine.
In a video released by the group, the officer said that the incident happened while he was drunk, and that he had opened fire due to his dislike for Wagner.