An Egyptian health ministry official, Khaled el-Khatib, was quoted in the media as saying that at least 34 people were killed and hundreds injured in the firing outside the building, where Morsi is said to have been put "under guard".
President Morsi, 61, was toppled by the powerful military on Wednesday and kept under detention, along with some senior aides of his Muslim Brotherhood party.
Meanwhile, in a statement, the Egyptian army said, "At dawn, an armed terrorist group tried to storm the Republican Guard (building), attacking army troops and police."
One army officer had been killed and 40 wounded, it said.
The army arrested 200 attackers who were carrying guns, ammunition and Molotov cocktails, according to the state-run MENA news agency.
The Muslim Brotherhood contradicted the army's statement, saying military opened fired on its supporters, killing several pro-Morsi protesters.
Angry over the incident, the group's political wing, the Freedom and Justice Party, called for "an uprising by the great people of Egypt against those trying to steal their revolution with tanks", media reports said.
It urged "the international community and international groups and all the free people of the world to intervene to stop further massacres and prevent a new Syria."
At least 36 people were killed and more than 1,000 wounded in the violence during pro-Morsi protests on Friday. The military has deployed troops in Cairo and other locations.
Supporters and opponents of the ousted president are regularly organising rival demonstrations, with the former demanding Morsi's reinstatement and the latter supporting Morsi's overthrow.
The Brotherhood-led National Alliance in Support of Electoral Legitimacy has vowed to hold protests in support of the deposed president.
Image: Members of the Muslim Brotherhood and supporters of ousted Egyptian President Mohamed Morsi run during a clash with anti-Mursi protesters | Photograph: Louafi Larbi/Reuters
Dramatic photos: Clashes grip Egypt as Islamists split over Morsi
PHOTOS: Egypt on verge of second bloody revolution
Cairo Diary: Arrest and freedom in new Egypt
Dramatic photos: Clashes grip Egypt as Islamists split
Morsi had to go to avert civil war in Egypt: ElBaradei