Abu Ismail, a popular Salafist figure who was a front-runner in the 2012 presidential race before being disqualified, was moved to Tora prison, eastern Cairo, the source added.
The Egyptian military has warned over any "provocative acts" against peaceful protesters.
"The right to peaceful demonstration is guaranteed to all and the army is trying to protect all sides," according to a statement by the military spokesman obtained by the Anadolu Agency.
The statement dismissed as baseless reports that the Armed Forces had asked people to refrain from protesting to facilitate its efforts to impose law and order in the streets.
Tens of thousands of ousted president Mohamed Morsi's supporters have taken to the streets in capital Cairo and other governorates to protest his overthrow, demanding his reinstatement.
One killed in clashes between Morsi supporters, opponents in Alexandria
At least one person was killed in clashes between supporters and opponents of ousted President Mohammad Morsi in the coastal city of Alexandria, medical sources told the Anadolu Agency.
Tens of thousands of Morsi loyalists have taken to the streets in capital Cairo and other governorates to protest his overthrow under an army roadmap for the future of Egypt, demanding his reinstatement.