There is fighting in an around the city, the US-based Institute for the Study of War (ISW) said, warning that with Ukrainian supply routes were narrowing.
"The Russians may have intended to encircle Ukrainian forces in Bakhmut, but the Ukrainian command has signalled that it will likely withdraw rather than risk an encirclement," ISW said Saturday.
Ukraine and Russia have since the summer fiercely fought for the city, whose symbolic importance surpassed its military significance.
Pro-russian separatists in the Donetsk region posted a video purporting to show Wagner fighters in the suburbs north of Bakhmut, having taken control over the Stupki railway station.
Wagner, a private army headed by Yevgeny Prigozhin, has taken centre stage in the fight for the city, which has exposed rivalries with Russia's conventional forces.
On Friday, Prigozhin said his fighters had "practically encircled" Bakhmut, and only one road remained under Ukrainian control.
Prigozhin has for weeks been publicising the advances of his men towards the eastern city.
Prigozhin regularly posts videos of himself alongside mercenaries, on the ground or even in a fighter jet, in contrast with Russian generals criticised for shirking the frontline.
In a rare exception, Russia on Saturday released a video of defence minister Sergei Shoigu inspecting troops in frontline regions in Ukraine.
The ministry said Sergei Shoigu inspected an advance command post in the South Donetsk direction without specifying exactly where or when.
He was seen travelling in a helicopter and talking to a soldier in front of damaged buildings.
The ISW think tank said Shoigu went there "likely to assess the extent of Russian losses around Vugledar and the possibility of a further offensive in this direction".
While the epicentre of the fighting is in the east of Ukraine, the death toll from a strike this week on an apartment block in southern Zaporizhzhia has now risen to 13.
Zaporizhzhia is one of the four regions -- along with Donetsk, Lugansk and Kherson -- that Russia claims to have annexed but never fully controlled.
Still, Moscow's forces have held the Zaporizhzhia nuclear power plant since March 4, 2022.
The plant has repeatedly made headlines and revived fears of nuclear catastrophes similar to the deadly Chernobyl disaster that shook Ukraine in 1986.
The exiled mayor of Energodar, which houses the central, told AFP Russia uses the plant as a "nuclear shield" to for its troops and equipment.
Kyiv and Moscow have blamed each other for shelling around the plant and the International Atomic Energy Agency (IAEA) posted observers there.
Ukrainian president Volodymyr Zelensky said on Saturday that Russia took the nuclear power plant "hostage" a year ago, and "turned the territory of the (power plant) into a de facto military training ground".
Zelensky on Saturday also met with European Parliament's president Roberta Metsola, who urged for Ukraine to be allowed to begin EU membership negotiations this year.
The United States announced a new $400 million security aid package for Ukraine on Friday that featured a range of ammunition for Kyiv's forces.