The Ukrainian army on Sunday said it retained control of an "insignificant" part of Bakhmut, the eastern city that Russia claims to have captured, but was advancing on its flanks.

A day earlier Russia's Wagner mercenary group and regular army claimed to have fully taken control of Bakhmut, scene of the longest and bloodiest battle in Moscow's invasion.

"Despite the fact that we now control an insignificant part of Bakhmut, the importance of its defence does not lose its relevance," the commander of Ukraine's ground forces, Oleksandr Syrsky, said.

"We continue to advance on the flanks in the suburbs of Bakhmut."

Earlier Ukrainian President Volodymyr Zelensky told the G7 summit in Hiroshima that Bakhmut was "not occupied".

Syrsky said the army would continue to defend Bakhmut. 

He called the situation for Ukrainian troops there "difficult but under control."

The chief of Wagner -- whose fighters have spearheaded Russia's advance on Bakhmut -- insisted there were no Ukrainian troops left in Bakhmut. 

"There is not a single Ukrainian soldier in Bakhmut as we have stopped taking prisoners," Wagner chief Yevgeny Prigozhin said in a post on Telegram.

"There are a huge number of corpses of Ukrainian soldiers."

He said that Bakhmut has been "taken to the last centimetre." 

Prigozhin said that Zelensky was either not telling the truth or "like many of our own military leaders, simply does not know what is happening on the ground, this is a possibility."

Both sides are believed to have suffered huge losses in the battle for Bakhmut.