“Wagner units have taken control of the whole territory of Soledar,” Prigozhin said through his press service on social media, adding that there was fighting in the centre of the city.
The extent of Wagner’s control over the city was unclear and AFP was not able to verify the claim.
The Kremlin-linked businessman said the “number of captives” will be announced Wednesday.
Ukraine has said its army was holding out in Soledar, a salt mining town in the Donetsk region around 15 kilometres (nine miles) from Bakhmut, which Russia has been trying to seize for months.
Both sides have said battles for Soledar have been bloody and intense.
Prigozhin published a photo of himself in military uniform surrounded by posing Wagner fighters, but did not say where it was taken.
Russian state news agency RIA Novosti published another photograph of Prigozhin with armed fighters, saying it had been taken in the salt mines of Soledar.
Prigozhin has said that Soledar has been stormed “exclusively” by Wagner units.
His and Wagner’s influence in Russia has risen significantly since the Kremlin sent troops to Ukraine in February last year.
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