People fought with rocks, firebombs and sticks in large-scale clashes that marked the worst violence of a deepening crisis over the disputed constitution.
The country's Health Ministry said 126 people were wounded in the fights, that were still raging hours after nightfall.
Three of Morsi's aides have resigned in protest of his handling of the crisis. This is in addition to two aides who had quit earlier. Now, five of his panel of 17 advisers have left their posts since the problem began.
A leading opposition advocate of reform and democracy said Morsi's rule was no different from that of former President Hosni Mubarak, whose authoritarian regime was toppled in an uprising nearly two years ago.
The opposition is demanding that Morsi take back the decrees giving him nearly unrestricted powers and shelve a disputed draft constitution that the president's Islamist allies passed last week in a rush.
The violence from the dueling sides is part of a political crisis that has divided Egypt into two camps: Islamists versus an opposition made up of youth groups, liberal parties and large sectors of the public. Both sides are refusing to give in, signaling a prolonged standoff.
The latest clashes started when thousands of Islamist Morsi supporters descended on the area around the palace where nearly 300 of his opponents were having a sit-in. The Islamists chased the protesters away from their base outside the palace's main gate and tore down their tents.
The protesters scattered to side streets, where they chanted anti-Morsi slogans. After a lull in fighting, hundreds of Morsi opponents arrived at the scene and began throwing firebombs at the president's backers, who responded with rocks.
By nightfall, there were about 10,000 Islamists outside the palace. Some of them appeared to plan staging their own sit-in.
Tuesday, at least 100,000 opposition supporters rallied outside the palace, and smaller protests were staged by the opposition elsewhere in Cairo and across much of Egypt. It was the latest of a series of mass protests against the president.
Morsi supporters have been calling on the opposition to enter a dialogue with the Islamist leader. But the opposition says a dialogue is useless unless the president first rescinds his decrees and shelves the draft charter.
The Associated Press contributed to this report.