The millionaire and longstanding Vladimir Putin ally said in a video published on Telegram that the handover will be completed by 1 June.
There was no immediate comment from the Russian defence ministry.
It was not possible to verify whether Wagner forces have begun to pull out of the heavily damaged city.
On Wednesday, the Ukrainian General Staff said heavy fighting was continuing inside Bakhmut after a nine-month battle, which has killed tens of thousands of people.
More on Russia
Related Topics:
Ukraine’s deputy defence minister said on Thursday that Wagner units have been replaced by regular troops in the suburbs but Wagner fighters remain in the centre of the city.
Ukrainian forces still have a foothold in the southwestern outskirts, Hanna Maliar said.
Advertisement
A Wagner victory in Bakhmut would be a much-needed boost for Mr Putin after his invasion lost momentum.
Top Ukrainian presidential adviser Mykhailo Podolyak said Ukraine’s long-awaited counteroffensive is already under way, saying it should not be anticipated as a “single event” starting “at a specific hour of a specific day”.
He said on Twitter that “dozens of different actions to destroy Russian occupation forces” were “taking place yesterday, are taking place today and will continue tomorrow”.
Mr Prigozhin has been in a long-running feud with Moscow and, during the 15-month war, he has repeatedly chastised Russia’s military leadership, accusing them of incompetence and failure to properly supply his troops.
Wagner’s involvement in the capture of Bakhmut has upped Mr Prigozhin’s standing, which he has used to set out his personal views about the war.
“Prigozhin is … using the perception that Wagner is responsible for the capture of Bakhmut to advocate for a preposterous level of influence over the Russian war effort in Ukraine,” according to the Institute for the Study of War.
Please use Chrome browser for a more accessible video player
3:46
Analysis: Wagner boss appears to insult Putin
His frequent critical commentary about Russia’s military performance is unusual under Mr Putin’s censorious regime.
His latest statement on Bakhmut came a day after he again broke with the Kremlin line on Ukraine.
He said its goal of demilitarising the country has backfired, acknowledged Russian troops have killed civilians and agreed with western estimates that he lost more than 20,000 men in the battle for the city.