The source from the pro-Morsi National Alliance for the Defense of Legitimacy said the 23-year-old victim had sustained birdshot wounds in the chest and abdomen.
Other demonstrators were injured during the protest dispersal, said the source, who asked not to be named, without giving a specific number.
The Anadolu Agency could not immediately obtain comments from the Interior Ministry on the alleged fatality.
Mohamed Sultan, head of Egypt's ambulance service, meanwhile, said his agency had not yet received any reports of fatalities.
The ousted president's supporters staged protests in different parts of the country on Friday in response to calls by the pro-Morsi alliance, which consists of a handful of pro-democracy groups and Islamist parties.
The alliance had called on supporters to stage protests on Friday under the slogan, "Together we'll complete our revolution."
The Interior Ministry says it disperses unlicensed protests in line with a law that makes it necessary for protest organizers to obtain prior authorization from the authorities.
Morsi backers consider his ouster by the army last year as a "military coup."
His opponents say his ouster was in response to mass protests that erupted against his government.