Yevgeny Prigozhin, the founder of Wagner, said he had taken over Bakhmut on Saturday, but Kyiv says it still holds parts of the city.
Ukraine says its troops are still making progress on the outskirts of Bakhmut.
But Mr. Prigozhin said his troops would start turning the city over to the Russian army on Thursday.
“Wagner will leave Artemovsk from May 25 to June 1,” Mr. Prigozhin said in an audio recording sent on Telegram.
Bakhmut used to be called Artemovsk after a Soviet rebel, but Ukraine changed the name.
He said that Wagner had set up “defence lines” on the city’s west side before the move.
But Ukraine’s Deputy Defense Minister Hanna Maliar said that her country’s troops still have a small foothold inside the city and are advancing on the city’s outskirts. However, their “intensity” of movement had decreased.
She later wrote in a post on Telegram that Ukrainian troops still had control over “certain private facilities and the private sector in the ‘Litak’ area” of the city.
Analysts say that Bakhmut is not very important to Moscow from a strategic point of view, but that capturing it would be a symbolic win for Russia after the longest fight of the war in Ukraine so far.
Wagner’s mercenaries have been focusing on the city for months, and their expensive strategy of sending in waves of men over and over again seems to have slowly broken down Kyiv’s opposition.
In the past few days, the two sides have made comments about Bakhmut that are at odds with each other.
During a Sunday speech at the G7 meeting in Japan, the president of Ukraine, Volodymyr Zelensky, said that Bakhmut “is not occupied” by Russia.
After Wagner said it had taken the city, Vladimir Putin, the leader of Russia, gave Wagner his congratulations. Mr. Prigozhin noted this in a video shared on social media on Saturday. He was in the video with some of his soldiers.
The BBC Verify team has been looking at new videos from Bakhmut shared online by both sides.
Four videos shared online in the past few days are shown on the map, but we don’t know when they were shot.
In a film posted on Sunday, a Wagner mercenary flag is seen being raised on land to the west of the city that Russia just recently claimed. In another, Mr. Prigozhin is seen speaking from a place we’ve found to be near the city’s central train station.
Two more videos show Ukrainian troops at work in the city’s center and to the west, where they have been fighting to get back control.
