Russia's Wagner Group claimed the “legal” capture of the city hall in Ukraine's eastern city of Bakhmut on Monday.
But the claims were denied by Ukraine, which said its forces still held the city.
“This is the Wagner private military company, these are the guys who took Bakhmut. In a legal sense, it's ours,” Mr Prigozhin said.
Why is Bakhmut important?
The battle for Bakhmut has raged for months, with Ukraine warning that its fall would lead to the loss of huge chunks of its territory.
Ukrainian military leaders said on Monday that enemy troops had tried to take control of the city, but their forces had “repelled more than 20 enemy attacks”.
Wagner's claim came a day after President Volodymyr Zelenskyy praised Ukrainian troops' defence of the city, much of which now lies in ruins.
“I am grateful to our warriors who are fighting near Avdiivka, Maryinka, near Bakhmut … Especially Bakhmut! It's especially hot there today!” Mr Zelenskyy said on Telegram.
In Kostyantynivka, a town about 27km from Bakhmut, a “massive attack” of Russian missiles on Sunday killed three men and three women, leaving another 11 wounded, Ukrainian authorities said.
The attack targeted residential areas where “ordinary civilians” were living, Mr Zelenskyy said.
There was a large crater in a yard and windows were shattered from ground to top floors in two 14-storey tower blocks, while private homes near by had smashed roofs, AFP said.
Liliya, a 19-year-old psychology student, stood outside her severely damaged high-rise block.
“Everything is bombed out. And I think it's like that in every flat, in fact. Because it was such an impact that it was very hard for anything to stay unbroken,” she told AFP.
What is Wagner group?
The Wagner Group is a mercenary force providing fighters for hire that was founded by Mr Prigozhin, an ally of Russian President Vladimir Putin
Wagner has become a key component of the Ukraine campaign and “almost certainly” commands 50,000 fighters in Ukraine, the UK Ministry of Defence has said.
In February, the White House put the toll of Wagner fighters killed or injured at 30,000.
The group has pulled inmates from prisons and sent them to the front lines, the Human Rights Council-appointed analysts said.
Who was Vladlen Tatarsky?
The number of people wounded in the bomb blast that killed Russian military blogger Vladlen Tatarsky in St Petersburg on Sunday has risen to 32, Russia's RIA state media said, citing the Ministry of Health.
Mr Tatarsky was killed in a St Petersburg cafe in what appeared to be the second assassination on Russian soil of a figure closely associated with the war in Ukraine.
It is not immediately known who was behind the killing. Russia's state Investigative Committee said it had opened a murder investigation.
RIA said on Monday that 10 of the people wounded were in a serious condition.
Russia's Foreign Ministry paid tribute to Tatarsky, calling him as one of the “defenders of the truth” and lashing out at Western governments for failing to react to the attack.
“Russian journalists constantly feel the threat of reprisals from the Kyiv regime,” spokeswoman Maria Zakharova said on Telegram.
Moscow on Sunday condemned the western “hype” around the arrest of US journalist Evan Gershkovich on espionage charges, dismissing Washington's latest call for his release.
Russia's Foreign Minister Sergey Lavrov told his US counterpart Antony Blinken that Gershkovich had been “trying to receive secret information” when he was arrested last week.
Gershkovich, a Wall Street Journal reporter, is believed to be the first foreign journalist held for spying in post-Soviet Russia, and his arrest is expected to escalate the Kremlin's confrontation with the West.
His arrest on March 30 has drawn outrage from the West and is being seen as a serious escalation of Moscow's sweeping crackdown on the media.
“The timing of the arrest looks like a calculated provocation to embarrass the US and intimidate the foreign press still working in Russia,” the Wall Street Journal's board of opinion editors said.
The White House has condemned the allegations as “ridiculous” and has asked Americans to leave Russia for their safety.