The military unit did not say what had happened to Baghdadi, give further details or identify its sources.
The US-led coalition against Islamic State said on Friday it had no "releasable information" on Baghdadi's whereabouts.
Syria's army declared victory over Islamic State on Thursday, saying its capture of the jihadists' last town in the country marked the collapse of their three-year rule in the region.
But the army and its allies are still fighting Islamic State in desert areas close to Albu Kamal near the border with Iraq, the Syrian army said on Thursday.
On Friday, the Syrian Observatory for Human Rights said Islamic State has taken back control of half of Albu Kamal.
The capture of the border town had sealed "the fall of the terrorist Daesh organization's project in the region," an army statement said on Thursday, using the Arabic acronym for Islamic State.
An audio message purported to be from Baghdadi was released in September. Baghdadi declared himself caliph and heir to Islam's historic leaders from the great medieval mosque in Iraq's Mosul in 2014.