Moscow's opposition fighters claim to have captured two Russian soldiers in Belgorod, close to the Ukrainian border.
The top official of Belgorod claimed that he had agreed to meet the soldiers' captors if they were still alive.
Despite Kiev's denials, Russia has accused Ukraine of being directly responsible for recent attacks in its border regions.
On Sunday, the Russian army claimed it had struck a "terrorist" group close to the village of Novaya Tavolzhanka with artillery.
The statement also stated that "the enemy dispersed and fled.".
The two men would be turned over if Belgorod governor Vyacheslav Gladkov agreed to negotiations, according to a paramilitary group's earlier Telegram message.
The BBC has been unable to independently confirm the identities of the two captives who were allegedly shown in the video.
Described as a joint statement with the Liberty of Russia Legion (FRL), the message was published by the Russian Volunteer Corps (RDK).
Both groups support overthrowing President Vladimir Putin and reject his massive invasion of Ukraine that began in February of last year.
According to Ukrainian officials, the two paramilitary groups are made up of Russian citizens who desire to establish a "security zone" for Ukrainians.
Due to a cross-border raid in Bryansk, Russia, the RDK gained notoriety in March. Its founder is a Russian nationalist who is said to have ties to neo-Nazis.
The FRL is regarded as a distinct type of group that fights alongside Ukrainian forces in opposition to Russian forces.
In his response via Telegram video, Mr. Gladkov stated that he had consented to the talks if the soldiers could be proven to be alive but added that he believed they had most likely been killed.
In spite of calling the combatants in question "scoundrels, murderers, and fascists," he pledged to "guarantee safety" if the talks went forward.
Even though they asked him to meet them in Novaya Tavolzhanka, he refused, saying it would be too risky and that he would instead wait for them at a checkpoint in Shebekino.
It was unclear right away if there had been any head-to-head combat. Additionally, according to Mr. Gladkov, there was fighting with a group of "Ukrainian saboteurs.".
There have been a number of recent attacks in Belgorod, most notably a significant cross-border incursion late last month that, according to Moscow, resulted in the deaths of 70 insurgents.
Kyiv has denied having any involvement in the attacks directly.
It has, however, portrayed the escalating violence in Russian territory as an inevitable result of Russia's invasion last year.
