Russian invasion: Wagner fighters claim to capture two more blocks in Bakhmut
Reuters could not independently verify Russia's claim.
Britain said in an intelligence update on Friday that Ukrainian troops had been forced to withdraw from some territory in the city in the face of a renewed Russian assault.
Russian troops have been battling since last summer to capture the town in eastern Ukraine, which has taken on huge symbolic importance even though analysts say it has little strategic value.
Kyiv has said the battle for the town is key to holding back Russian forces along the entire eastern front.
"Wagner assault units have successfully advanced, capturing two districts on the northern and southern outskirts of the city," Russia's defence ministry said in a briefing on Saturday.
The Russian Wagner mercenary group, headed by Kremlin-linked businessman Yevgeny Prigozhin, has spearheaded much of the fighting for the city.