The suspect is a Syrian national and lives in an asylum-seeker shelter in the northeast region of Brandenburg, the newspaper reported, quoting security sources.
The man had allegedly told other migrants living in the centre that he had fought for the IS group and killed people.
His claims were filmed secretly with a mobile telephone by other residents of the shelter, leading to the probe.
Investigators are now trying to determine if the Syrian was indeed a member of the IS, the newspaper reported.
A spokeswoman for federal police declined to confirm or deny the report.
Despite the case, federal police do not believe that Islamists are infiltrating Germany through the influx of refugees.