Nine were injured outside the Ettehadiya Palace, along with two in Tahrir Square, during protests Monday commemorating former President Hosni Mubarak's downfall.
Several political parties and activist groups had called for the protests to demand President Mohamed Morsy's resignation and an end to the growing influence of the Muslim Brotherhood.
Nine were injured by tear gas at the palace during clashes between some protesters and security forces, according to Mohamed Sultan, director of Egypt's ambulance service.
Sultan added that two people were injured in Tahrir after clashes between passengers and protesters who tried to block trains at the Sadat metro station. However, he said the two injured only suffered mild bruises.
A media official at the Interior Ministry had said that backup security forces were sent to the area near Ettehadiya Palace after some demonstrators attempted to attack the building.
Edited translation from DPA