The battle for Bakhmut has turned into one of the bloodiest of the 14-month war, with the eastern Ukrainian city almost completely destroyed by artillery shelling and urban combat.
Russia says capturing Bakhmut will allow it to mount further offensives into eastern Ukraine.
If it succeeds, Moscow's forces are likely to face even larger urban battles for the nearby towns of Kramatorsk and Sloviansk.
Russia's defence ministry on Sunday said troops secured two more blocks in Bakhmut's western districts, with airborne units providing reinforcements to the north and south.
Kyiv has repeatedly denied claims its forces are poised to withdraw.
Reuters was unable to verify battlefield reports.
Also on Sunday, the Russian-installed head of Ukraine's southern Kherson region denied a report by a US think tank that Ukrainian forces had taken up positions on the eastern bank of the Dnipro River.
"There is no enemy foothold on the left (eastern) bank of the Dnipro River ... our military completely controls that territory," Vladimir Saldo wrote on his Telegram channel.
"There may be cases of enemy sabotage groups making landings to take a selfie, before being ... destroyed or pushed into the water by our fighters."
Citing Russian military bloggers embedded with Moscow's forces, the Institute for the Study of War said Ukraine had "established positions" on the eastern bank, although it was not clear "at what scale or with what intentions".
A representative for Ukraine's southern command neither confirmed nor denied the report and called for "informational silence" to ensure operational security.
"I want everyone to understand that it is very difficult work to cross an obstacle like the Dnipro, for example, when the front line runs along such a wide, powerful river," Natalia Humeniuk told Ukrainian television.
"It's necessary to gather up some patience," she said.
The reports came as former Russian president Dmitry Medvedev said Moscow would terminate the Black Sea grain deal that enables vital exports from Ukraine if the G7 moves to ban exports to Russia.
The Group of Seven (G7) countries are considering a near-total ban on exports to Russia, Japan's Kyodo news agency reported last week, citing Japanese government sources.
Russia has repeatedly threatened to scrap its participation in the grain deal, due to expire on May 18.
"This idea from the idiots at the G7 about a total ban of exports to our country by default is beautiful in that it implies a reciprocal ban on imports from our country, including categories of goods that are the most sensitive for the G7," Medvedev said in a post on his Telegram channel.
"In such a case, the grain deal - and many other things that they need - will end for them," he said.
The G7 is reportedly discussing reversing its sanctions approach so exports to Russia are automatically banned unless they are included on a designated list of products allowed to be shipped to the country.
Under the current framework, goods are allowed to be sold to Russia unless they are explicitly black-listed.
Medvedev, a long-time ally of Russian President Vladimir Putin, is Putin's deputy chair at the influential Security Council and heads a government commission on arms production for the war in Ukraine.
The G7 called on Sunday for the "extension, full implementation and expansion" of the deal to export Ukrainian grain through the Black Sea, the group's agriculture ministers said in a communique.
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