The Islamist President, Mohamed Mursi, criticised by his opponents for his silence in the last few days, was due to address the nation later in the day, state television said.
Hundreds of his supporters who had camped out near the palace overnight withdrew before a mid-afternoon deadline set by the Republican Guard.
Dozens of Mr Mursi's foes remained, but were kept away by a barbed wire barricade guarded by tanks.
Mr Mursi's Islamist partisans fought opposition protesters well into the early hours during duelling demonstrations over the President's decree last month to expand his powers to help him push through a mostly Islamist-drafted constitution.
Officials said seven people had been killed and 350 wounded in the violence. Six of the dead were Mursi supporters, the Muslim Brotherhood said.
The commander of the Republican Guard said the deployment of tanks and troop carriers around the presidential palace was intended to separate the adversaries, not to repress them.
General Mohamed Zaki said: "The armed forces, and at the forefront of them the Republican Guard, will not be used as a tool to oppress the demonstrators."
Hussein Abdel Ghani, spokesman for the opposition National Salvation Front, said more protests were planned, but not necessarily at the palace in Cairo's Heliopolis district.
Egypt plunged into renewed turmoil after Mr Mursi issued his decree and an Islamist-dominated assembly hastily approved a new constitution to be put to a referendum on December 15.
The Supreme Guide of the Brotherhood, to which Mr Mursi belonged before he was narrowly elected president in June, appealed for unity.
Rival factions used rocks, petrol bombs and guns in the clashes around the presidential palace.
Demonstrator Emad Abou Salem, 40, said: "We came here to support President Mursi and his decisions. He has legitimacy and nobody else does."
Mr Mursi's opponents accuse him of seeking to create a new "dictatorship". The President said his actions were necessary to prevent courts still full of judges appointed by Mubarak from derailing the constitution.
Mr Mursi has shown no sign of buckling under pressure from protesters, confident the Islamists, who have dominated both elections since Mubarak was overthrown, can win the referendum and the election to follow.
Mahmoud Hussein, the Brotherhood's secretary-general, said holding the vote was the only way out of the crisis, dismissing the opposition as "remnants of the [Mubarak] regime, thugs and people working for foreign agendas".
As well as relying on his Brotherhood power base, Mr Mursi may also tap into a popular yearning for stability and economic revival after almost two years of political turmoil.
l Egypt's Government must protect peaceful protesters and prosecute anyone inciting violence, including politicians, UN Human Rights Commissioner Navi Pillay has said.
She said: "People have the right to protest peacefully and they have the right not to be killed or injured for doing so."
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