Rustam Gelayev was killed in the shelling of a mosque in Aleppo by forces loyal to Assad earlier this month, according to the website chechenews.com.
The report, which cited unidentified sources in Chechnya, said Gelayev, 24, had joined a unit of ethnic Chechen volunteers fighting alongside Syrian insurgents in a 17-month-old uprising against Assad, who has close ties with Moscow.
Another website sympathetic to the insurgents, kavkazcenter.com, said he had been killed when his unit "entered into a battle with superior forces of the Alawite regime ... presumably between Aug. 11 and Aug. 13."
Syria's conflict pits largely Sunni Muslim insurgents against Assad's minority Alawite community that has long dominated the power structure. Chechens are Sunni Muslims.
The Russian newspaper Kommersant, however, cited a relative of Gelayev as saying he had been studying in Syria, had decided to leave because of the violence and was making his way to Turkey when he was killed.
There is a large ethnic Chechen diaspora in Turkey, whose prime minister, Tayyip Erdogan, was once an Assad ally but is now among his most vocal critics.
Russia, which has shielded Assad from concerted international pressure to end the bloodshed by vetoing three Western-backed U.N. Security Council resolutions, says some of those fighting the government are foreign "terrorists".
A Russian diplomat said in March that there were at last 15,000 foreign fighters battling Assad's government.
Chechnya's Kremlin-backed regional leader, Ramzan Kadyrov, said on Thursday that "no Russian citizens of Chechen nationality are fighting in Syria on any side", but indicated insurgents or Chechens living abroad could be doing so.
"Chechens are not fighting in Syria - at least we can very responsibly say that about residents of the Chechen Republic," Kadyrov told reporters. "As for Gelayev ... and others like him, they have no homeland - they lost it long ago."
The body of Gelayev was brought to Chechnya and he was buried there in his ancestral village, the reports said.
His father, Ruslan Gelayev, was a prominent warlord who fought Russian federal forces in the two post-Soviet separatist wars in the 1990s and early 2000s in Chechnya, one of the mostly Muslim provinces of Russia's North Caucasus.
Ruslan Gelayev was fatally wounded in a gunbattle with Russian border guards on the frontier with Georgia in 2004.
Relatives of Gelayev and Chechen government officials could not immediately be reached for comment.
Russian media say Gelayev's mother is an ethnic Russian living outside Chechnya and that he spent little time there. Kommersant said he had moved to Belgium with his wife and child, then to Egypt several years ago to study Islam.