Ukraine has called Moscow's claims that its fighters crossed into Russia and killed two civilians a "deliberate provocation".
If confirmed, the alleged incursion would be a rare instance of fighting inside Russian territory, despite its claims that regions on its border are routinely shelled by Ukrainian forces.
The "Ukrainian nationalists" had later been pushed back over the border and hit with a "massive artillery strike", Russia's FSB security service said.
President Vladimir Putin said the attackers, who he called "neo-Nazis and terrorists", had "penetrated the border area and opened fire on civilians".
But Ukrainian presidential adviser Mykhaylo Podolyak quickly dismissed Moscow's claims.
"The story about a sabotage group in Russia is a classic deliberate provocation," Mr Podolyak said.
"Russia wants to scare its people to justify the attack on another country and the growing poverty after the year of war."
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AFP could not independently verify the reports.
FSB said the incident happened in the Klimovsky district of the Bryansk region, which borders Ukraine, and that a "large number of explosives" had been found.
Earlier in the day, Bryansk regional Governor Alexander Bogomaz said "a reconnaissance and sabotage group" had crossed the border near the village of Lyubechane.
Mr Bogomaz said that two people were killed and a child wounded in the attacks.
He said the boy had been taken to hospital and had a bullet removed during surgery.
But a different account was put forward in two videos on social media, which showed four men in military uniform claiming to be from a Russian volunteer group in the Ukrainian army.
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The men said they were in Russia and made statements against the Kremlin, denying reports that they had taken hostages or killed anyone.
Mr Bogomaz also said a Ukrainian drone had attacked the village of Sushany, leaving a residential home in flames, and that mortar fire had damaged two homes in Lomakovka, also in Bryansk.
Roman Starovoyt, Governor of another Russian border region, said Ukrainian shelling on the village of Tetkino killed one person and injured another.
Mr Starovoyt said the shelling damaged three houses and cut power in the village.
In Ukraine, officials said a Russian strike on an apartment block in the southern city of Zaporizhzhia had killed at least three people.
"The terrorist state wants to turn every day for our people into a day of terror," Ukrainian President Volodymyr Zelenskyy said.
"But evil will not reign in our land. We will drive all the occupiers out and they will definitely be held accountable for everything."
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Ukraine tension disrupted the G20 meeting in India, with Russian Foreign Minister Sergey Lavrov telling the assembled delegates that western representatives had derailed the meeting, calling it a "farce".
US Secretary of State Antony Blinken said he had told Mr Lavrov to "end this war of aggression, engage in meaningful diplomacy".
Also on the diplomatic front, German Chancellor Olaf Scholz said that Berlin and its allies were in talks with Kyiv over future security guarantees in preparation for a sustainable peace for Ukraine.
"We are speaking with Kyiv and other partners over future security guarantees for Ukraine," he told the German Parliament.
"Such security guarantees, however, come with the presumption that Ukraine successfully defends itself in this war."
Mr Scholz will travel to the US for talks with President Joe Biden on Friday.
