Police fired teargas at demonstrators angered by the Islamist leader's drive to hold a referendum on a new constitution on December 15. Some broke through police lines and protested next to the palace's perimeter wall.
Several thousand people had gathered nearby in what they dubbed "last warning" protests against Mr Mursi, who infuriated opponents with a November 22 decree that expanded his powers. "The people want the downfall of the regime," the crowd chanted.
"The president left the palace," a presidential source, who declined to be named, said. A security source at the presidency also said the president had left the building.
Mr Mursi ignited a storm of unrest in his bid to prevent a judiciary still packed with appointees of ousted predecessor Hosni Mubarak from derailing a troubled political transition.
"Our marches are against tyranny and the void constitutional decree and we won't retract our position until our demands are met," said a spokesman for an opposition coalition of liberal, leftist and other disparate factions.
Opposition groups have accused Mr Mursi of making a dictatorial power grab to push through a constitution.
Cairo stocks closed up 3.5% yesterday as investors took heart at what they saw as prospects for a return to stability.
Mohamed Radwan, at Pharos Securities brokerage, said the Supreme Judicial Council's agreement to supervise the referendum had generated confidence that the vote would happen.