According to the Russian defense ministry, 250 Ukrainian soldiers were killed while a significant Ukrainian offensive was thwarted.
Kyiv hasn't commented, and no one else has independently verified the Russian claim.
Six mechanized and two tank battalions were used in Ukraine's offensive in the Donetsk region, according to the ministry.
Although a counter-offensive by Ukraine has been promised, Kyiv on Sunday requested silence before the operation.
Whether the alleged attacks signal the beginning of the offensive to retake Ukrainian territory from Russian forces is still unknown.
The enemy began a massive offensive in five sectors of the front on June 4 in the direction of South Donetsk, the Russian defense ministry reported on Telegram.
In what Kiev considered to be the most exposed area of the frontline, the ministry claimed that Ukrainians attempted to breach Russian defenses.
"The enemy failed in its attempts to complete its tasks. ".
Uploaded video showed aerial attacks on military vehicles. Moscow asserted that 16 tanks and 250 soldiers had been lost by Ukraine.
Ukraine has been preparing a counterattack for weeks. However, it has sought as much time as possible to prepare its soldiers and acquire military hardware from Western allies.
Officials in Kiev have issued a warning against public speculation about the offensive, claiming that it could aid the adversary.
Plans adore quiet. There won't be a start announcement, the defense ministry declared in a video shared on Telegram on Sunday.
The video showed masked, heavily armed soldiers pressing their lips with their fingers.
Much is at risk because the Ukrainian government must demonstrate to its citizens and Western allies that it is capable of breaching Russian defenses, ending the military standoff, and reclaiming some of its sovereign territory.
In another location, near the Ukrainian border in Belgorod, fighters opposed to the Moscow government claim to have captured some Russian soldiers.
The Liberty of Russia Legion (FRL), which made the assertion, claimed that the announcement was a joint one with the Russian Volunteer Corps (RDK).
Vladimir Putin's government is being challenged by both groups. They reject his February 2017 launch of a full-scale invasion of Ukraine.
Vyacheslav Gladkov, the top official in Belgorod, responded by stating that he had consented to meet the soldiers' captors if they were still alive. Later on though, the fighters claimed that the governor "had not found the courage" to meet them and that they would be turning over their prisoners to Ukraine.
Despite Kiev's denials, Russia has accused Ukraine of being directly responsible for recent attacks in its border regions.
Later, Mr. Gladkov claimed that a drone attack had set fire to a power plant in Belgorod.
Although there has been no independent confirmation of the attack, Moscow claims Ukraine has regularly targeted the area with drone attacks.
