The governor, Vyacheslav Gladkov, said he was ready for talks to retrieve "our guys" from the saboteur fighters who say they have taken Russian soldiers prisoner in the Belgorod region.
It was the first time a Russian official had admitted to the capture of prisoners on Moscow's territory by pro-Ukraine forces during the more than 15 months of conflict.
The pro-Kyiv groups made up of Russians, the Russian Volunteer Corps and the Legion for Freedom of Russia, had earlier called on the Belgorod governor to meet fighters and retrieve captive soldiers.
"The only thing stopping me from negotiating with them is our guys that are in their hands, maybe they are already dead," Gladkov said.
He said fighting was ongoing in the border village of Novaya Tavolzhanka.
Ukraine has intensively shelled Russian settlements on the border this week, forcing thousands to flee to the regional hub of Belgorod.
"A sabotage group came in, there is combat in Novaya Tavolzhanka," Gladkov said.
"I hope they will all be destroyed."
Russia's army subsequently claimed to have repelled a "sabotage group of Ukrainian terrorists" seeking to cross the frontier near to the settlement.
"The enemy was hit by artillery. The enemy scattered and retreated," it said in a statement.
Fighting around Novaya Tavolzhanka follows last month's dramatic armed incursion from Ukraine into the Belgorod region which forced Moscow to use its artillery and air force on home soil.
The border breach was claimed by anti-Kremlin nationalists.
Gladkov earlier asked residents of the Shebekino border district to leave their homes amid the shelling.
Earlier, the nationalist groups published a video addressing the governor. It claimed to show three "prisoners" who their captors called "simple soldiers sent by your leadership to this war". One appeared to have been injured and was on a drip on a hospital bed.
They called on the governor to meet them in a church in Novaya Tavolzhanka on the Orthodox "Trinity" holiday.
In a sign fighting was intensifying, Ukrainian authorities said Russian shelling Sunday killed two women in the Kharkiv region in the town of Vovchansk, near the border with Russia's Belgorod region.
"As a result of enemy strikes, two civilian women -- aged 62 and 74 -- died," the Kharkiv region Prosecutor's Office said.
Vovchansk is around 30 kilometres (18 miles) south of Russia's Shebekino.
Russians fleeing border regions
In Belgorod, AFP saw volunteers handing humanitarian aid to Russians fleeing border regions.
Ukraine has consistently not claimed responsibility for attacks on Russian soil, but presidential advisor Mykhaylo Podolyak said Sunday the situation in the border areas "should be viewed as the future of Russia".
The attacks on the Belgorod region come as Kyiv says it is preparing a major counteroffensive against Russian forces.
In a video published Sunday, the Ukrainian army appeared to call on soldiers to stay silent about the plans and said there will be no announcement on the start of the long-awaited offensive.
Outside the Ukrainian city of Dnipro, 300 kilometres southwest of Belgorod, the body of a two-year-old girl was pulled from the rubble left by a Russian strike on a residential area.
Authorities said the mother of the girl who was identified as Yelyzaveta Pryhodka remained in intensive care.
The girl's father, who recovered her body according to an official, was at work when the building was hit.
Officials said 22 people were wounded in the strike.
Ukrainian President Volodymyr Zelensky said five children were wounded in the strike.
In his daily address to Ukrainians, he said three boys -- aged six, 11 and 15 -- were in hospital in a "bad condition."
Zelensky said in a tweet earlier Sunday that 500 children had died since the start of the conflict while cautioning the exact number of children injured would be difficult to determine.
"Many of them could have become famous scholars, artists, sports champions, contributing to Ukraine's history. They fell victim to the enemy's missiles and hate," he tweeted.
Moscow has this week intensified its aerial assaults on Ukraine.
Kyiv also said a Russian strike had hit an airfield in central Ukraine overnight, which Moscow appeared to confirm on Sunday, saying it had struck military air bases with long-range weapons.
Russia also unleashed a series of air assaults on Kyiv during the past week, including rare daytime strikes.