However there was no immediate sign that the government was readying to carry out its threat made yesterday to remove Islamist protest camps set up to demand Morsi's reinstatement following his ouster by the army on July 3.
Under a cloud of balloons, thousands of men, women and children attended the dawn prayer marking the end of the fasting month in the Rabaa al-Adawiya and Al-Nahda squares, where Morsi's supporters have been camped for weeks calling for his reinstatement.
But leaders of the Muslim Brotherhood from which Morsi hails called for rallies "until victory", raising fears of further bloody confrontations.
Giving warning of the looming showdown, government daily Al-Gumhuriya today ran the headlines "Last warning" and "The hour of battle approaches".
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Several thousand supporters of the ousted president also marched in other parts of Egypt, including in Alexandria, the country's second city, and in Qena in the south.
The warning to the protesters came after Egypt's interim presidency said yesterday that Western and Arab efforts to mediate an end to the country's political deadlock had failed.
"These efforts have not achieved the hoped for results," the presidency said, referring to mediation by Burns and EU envoy Bernardino Leon, who were among a string of diplomats who had travelled to Cairo.
The presidency said it "holds the Muslim Brotherhood completely responsible for the failure of these efforts, and for consequent events and developments relating to violations of the law and endangering public safety."
Egypt is deeply divided over Morsi's ouster, and violence in the margins of demonstrations between his supporters and opponents has killed more than 250 people -- mostly Morsi's backers -- since the end of June.
UN chief Ban Ki-moon urged the release of Morsi, who has been formally remanded in custody and is being held incommunicado at an undisclosed location.
Washington yesterday urged Egypt's military and political factions to resolve their differences through dialogue.
"We absolutely do not believe that the time for dialogue has passed. We will continue this conversation, and it certainly remains a priority of ours and obviously a priority of the EU and other officials around the world who've been involved," State Department spokeswoman Jen Psaki said.
Morsi himself is being held on suspicion of having collaborated with Palestinian militants to kill policemen and stage jail breaks during an early 2011 uprising against strongman Hosni Mubarak, while Morsi was in prison.
The Brotherhood's supreme guide, Mohamed Badie, and his deputies are to stand trial later this month on charges of inciting the shootings of protesters outside their headquarters on June 30.
The government says it is up to the judiciary to release prisoners. The Islamists say their release is a precondition for further talks on finding a settlement, which could include symbolically reinstating Morsi, who would then call early elections.
More than 80 protesters were killed in clashes with police at the main sit-in outside Cairo's Rabaa al-Adawiya mosque on June 27.
Over 50 people died in earlier clashes with soldiers outside an elite army base.