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.
According to the ministry, Ukrainian troops "while retreating, are deliberately destroying city infrastructure and residential buildings in order to slow the advance" of Moscow's forces.
AFP was unable to verify the situation on the ground.
Both Russia and Ukraine are believed to have suffered huge losses in the battle for Bakhmut.
Russia said Friday it was pushing to take the western districts of the heavily destroyed salt mining town that once had a population of 70,000 people.
On Thursday, Moscow claimed to have cut off Ukrainian forces in Bakhmut. Kyiv denied the claim, saying it had access to its troops and was able to send in munitions.
