Video of what Russia says is the battle appears to show military vehicles coming under heavy fire in fields.
But there has been no comment from Kyiv and Russia's claim has not been independently verified.
A Ukrainian counter-offensive has been long awaited but Kyiv says it will not give advance warning of its start.
It is too early to say whether this alleged offensive is the start of Kyiv's long-awaited counterpunch.
However, with Ukraine claiming to have made marginal gains elsewhere on the front line, there has been a notable increase in military activity.
The Russian defence ministry said Ukraine had launched the "large-scale offensive" in the Donetsk region on Sunday using six mechanised and two tank battalions.
It claimed the Ukrainians tried to break through Russian defences in what Kyiv saw as the most vulnerable part of the front line - but that it "did not achieve its tasks, it had no success".
Moscow claimed Ukraine had lost 250 troops as well as 16 tanks.
The footage featured masked and well-armed troops holding their fingers against their lips.
The claims by Russia's defence ministry are not yet verified. If the footage of armoured vehicles coming under heavy fire is authentic, then it reflects the stiff resistance Ukrainian forces will face as they try to liberate more territory.
And if it is not what it seems, it is still an attempt by Moscow to take control of the narrative.
There has been a significant increase in Ukrainian messaging on when and how their much-anticipated counter-offensive could take shape.
Ukraine has been planning a counter-offensive for months. But it has wanted as much time as possible to train troops and to receive military equipment from Western allies.
Officials in Kyiv have warned against public speculation over the offensive, saying it could help the enemy.
"Plans love silence. There will be no announcement of the start," the defence ministry said in a video posted to Telegram on Sunday. Its footage featured masked and well-armed troops holding their fingers against their lips.
It will take Ukraine time to achieve its goal of liberating territory taken by Russia as far back as nine years ago.
And Moscow has had time to prepare. It means if Ukraine is able to mount a counter-offensive, it is going to take a while.
Much is at stake because the government in Kyiv needs to show the people of Ukraine - and Western allies - that it can break through Russian lines, end the effective military deadlock and recapture some of its sovereign territory.
On Monday morning, the commander of Ukraine's ground forces, Oleksandr Syrskyi, said troops were "moving forward" towards Bakhmut and had destroyed a Russian position near the city.
Elsewhere, fighters opposed to the government in Moscow say they have captured some Russian soldiers in Belgorod, near the border with Ukraine.
The claim was made by the Liberty of Russia Legion (FRL), which described the announcement as a joint statement with the Russian Volunteer Corps (RDK).
Both groups want to topple President Vladimir Putin. They oppose the full-scale invasion of Ukraine that he launched in February last year.
Belgorod's top official, Vyacheslav Gladkov, replied to say he had agreed to meet the men's captors if the soldiers were still alive. But the fighters later said that the governor "had not found the courage" to meet them and they would hand over their captives to Ukraine.
Russia has blamed Ukraine for recent attacks in its border territories, but Kyiv denies being directly involved.
Authorities in Belgorod said an energy plant was ablaze following a drone attack on Monday morning.
And in Russia's Kaluga region - which borders the southern districts around Moscow - governor Vladislav Shapsha said two drones fell onto a main road. Mr Shapsha said there had not been an explosion and the area was now cordoned off.
There has been no independent confirmation of either attack, but Moscow says the Belgorod region has been the regular target of drone attacks from Ukraine.