The Wagner Group said in a post on Telegram that it had documents belonging to the men.
The Wagner Group said in a post on Telegram that it had documents belonging to the missing men – which it claimed it had found with the body of one of them.
It said he had died during a battle with the Wagner Group near Soledar.
The post said: "On January 8, the Ask Wagner hotline received a request to find two British citizens who disappeared on January 6 in Soledar - Andrew Bagshaw and Christopher Parry.
"Today the body of one of them was found, documents on both Britons were found with him."
Andrew Bagshaw, 48 and Christopher Parry, 28, are understood to have travelled to Ukraine to help with the civilian evacuation and humanitarian efforts, the Foreign Office said last week.
The Foreign Office, which has said it is supporting the families of the two men, has not confirmed the claim.
Andrew Bagshaw, 48 and Christopher Parry, 28, went to volunteer in Ukraine to help with the civilian evacuation and humanitarian efforts, the Foreign Office said last week.
Mr Parry, reportedly a running coach from Cheltenham, had previously said he had been driving to towns and villages on the frontline to evacuate local residents.
Meanwhile, Mr Bagshaw, a dual New Zealand/British national, is reportedly to have travelled to Ukraine from New Zealand and had been delivering aid and evacuating citizens.
The family of Mr Parry has previously spoken of their pride in his work but expressed concern for his health and whereabouts.
And the parents of Mr Bagshaw, Philip and Susan, previously confirmed their son had been reported missing in Ukraine and were said to be "very grateful for all the agencies from both London and New Zealand who are working so hard to find him".
A spokeswoman for the couple said they were "immensely proud" of his work delivering food and medicines and assisting elderly people to move from near the front line.
The Foreign Office is warning against all travel to Ukraine, saying there is "a real risk to life".