According to Deputy Defence Minister Hanna Maliar, Ukrainian forces have advanced around Bakhmut, identifying the eastern city as the “epicentre of hostilities.”
The Ukrainian government did not specify whether this was the start of a long-awaited counter-offensive.
Separately, Russia announced on Monday that it had repulsed a new onslaught in Donetsk.
Bakhmut has been at the center of fierce conflict for months. It has minimal military significance, but it is significant symbolically for both Kyiv and Moscow.
The claims made by Ukraine and Russia on Monday have not been independently verified.
Ukrainian military sources informed the BBC earlier in the day that a series of small-scale armoured offensive operations were underway.
Ms Maliar stated on social media that “despite stiff resistance and the enemy’s attempts to hold their positions, our military units advanced in multiple directions during the fighting.”
She stated that Ukrainian troops moved from 200m to 1,600m (656-5,250ft) in Orikhovo-Vasulivka and Paraskoviivka, while they advanced between 100m and 700m (330-2,300) in Ivanivske and Klishchiivka.
All four settlements are within a few kilometers of Bakhmut. The struggle for the city in the Donetsk region was the war’s longest and bloodiest.
President Volodymyr Zelensky congratulated Ukrainian fighters in a video speech late Monday for providing “the news we are expecting” on the Bakhmut route.
Wagner, a Russian paramilitary force, claimed to have taken control of the city in late May. Some commentators believe Kyiv’s forces are seeking to encircle Bakhmut and trap Russian units in recent weeks.
A big Ukrainian counter-offensive has been long anticipated, but Kiev has already stated that it will not provide early notice of its launch.
Military activity has increased noticeably, with Ukraine claiming to have achieved minor gains elsewhere on the front lines.
The newest findings are being interpreted as a new evidence that the anticipated Ukrainian drive has begun.
The Russian defense ministry announced on Monday that a new Ukrainian onslaught in Donetsk had been repulsed.
According to the report, the attacking side sustained tremendous casualties and lost 28 tanks, including eight German-made Leopards. The BBC has not independently verified this.
This comes a day after Moscow announced that a Ukrainian “large-scale offensive” in the Donetsk region had began on Sunday but had failed. Ukraine’s military claimed it has no knowledge of a big attack in the region.
Russia said that video of the battle in Donetsk showed military trucks under heavy fire in fields. Russia claims to have killed 300 soldiers and destroyed 16 tanks.
“We do not have such information and do not comment on any kind of fake,” a Ukrainian army official told Reuters.
Ukrainian messaging about when and how their counter-offensive would take shape has increased significantly.
Ukraine has been planning a move like this for months. However, it has sought as much time as possible to train troops and collect military supplies from Western allies.
According to Reuters, Foreign Minister Dmytro Kuleba claimed his country now has enough weapons for a counter-offensive but would not comment on whether it has begun.
Officials in Kyiv have warned that public speculation about the offensive could aid the adversary.
“Plans enjoy silence. “There will be no announcement of the start date,” the defense ministry said in a video aired lately. It showed masked and heavily armed troops with their fingers against their mouths.
It will take time for Ukraine to liberate territories occupied by Russia as recently as nine years ago.
And Moscow has had plenty of time to prepare. It means that even if Ukraine is able to mount a counter-offensive, it will take time.
Ukraine’s counter-offensive is’ready to launch.’
Who are the Ukrainian fighters penetrating Russia?
The mothers and their children are returning to the front lines.
Much is at stake because the Ukrainian government needs to demonstrate to its people – and Western partners – that it can break through Russian lines, end the effective military standoff, and reclaim some of its sovereign territory.
In other news, anti-government fighters in Moscow claim to have taken some Russian soldiers in Belgorod, near the Ukrainian border.
The Liberty of Russia Legion (FRL) made the claim, describing the news as a joint statement with the Russian Volunteer Corps (RDK).
Both factions aim to depose President Vladimir Putin and oppose his full-fledged invasion of Ukraine, which began in February of last year.
Russia has blamed Ukraine for recent strikes in its borderlands, but Kiev denies any involvement.
Meanwhile, on Monday, President Zelesnky met with UK Foreign Secretary James Cleverly in Kyiv.
Mr Zelensky stated that the two discussed the impending Nato summit in Vilnius in July, as well as Ukraine’s peace efforts.
