Moscow says capturing Bakhmut opens the way to advances in the eastern region known as the Donbas. Kyiv says the battle drew Russian forces into the city, inflicting high casualties and weakening Moscow's defensive line elsewhere.
Wagner founder Yevgeny Prighozin, who has repeatedly accused Russia's regular military of abandoning ground captured earlier by his men, said Wagner would be ready to return to the city if needed.
"From today at five in the morning, May 25 until June 1, most of the (Wagner) units will rebase to camps in the rear," Prighozin said in a video, wearing battle gear and standing beside a war-damaged residential block. He says 20,000 of his fighters died taking Bakhmut.
Ukrainian Deputy Defence Minister Hanna Maliar said in a message on the Telegram app that Wagner had handed over positions on the city's outskirts but "inside the city itself Wagner fighters remain".
The replacement of Wagner's forces with regular soldiers could provide a tempting target for Ukraine, after its advances on some higher ground around the city over the past week.
Serhiy Cherevatyi, a spokesperson for Ukraine's eastern military command, said the number of Russian attacks in the area had fallen in the last three days, and that there had been two military engagements in the last 24 hours, though shelling continued.
SHELLING, REGROUPING
"We can definitely note a reduction in attacks and possibly this is linked to their regrouping. It is clear that we have inflicted heavy losses and they need this (to regroup)," he said.
Maliar said Russia was also reinforcing its positions on the flanks of Bakhmut and shelling Ukrainian forces to try to stop the Ukrainian advances. Ukraine says it retains a small foothold within the city.
Ukraine is planning a major counteroffensive against Russia's invasion, using modern weapons supplied by Western allies and troops freshly trained across Europe. It has not said when the offensive will start of which territory it will seek to retake.
Russia has built sprawling fortifications in its neighbour's east and south in readiness, and military analysts say recent small Ukrainian offensives may be aimed at spreading Russia's forces thinly, creating gaps elsewhere that can be exploited.
Russian President Vladimir Putin says the United States and its allies are fighting an escalating proxy war against Russia after he sent troops to invade Ukraine in February last year.
The United States and its allies say they want Ukraine to defeat Russian forces on the battlefield but deny Putin's claims that they want to destroy Russia, which they accuse of an unprovoked, imperial land grab in Ukraine.
Russia moved ahead on Thursday with a plan to deploy tactical nuclear weapons in Belarus, signing a deal with its ally about the storage of the warheads at a special facility that should be finished in just over a month's time.
(Reporting by Reuters; Writing by Frank Jack Daniel; Editing by Andrew Heavens)
By Pavel Polityuk
