"We give Mohamed Morsi until 5pm on Tuesday July 2 to leave power, allowing state institutions to prepare for early presidential elections," the opposition movement Tamarod said in a statement on its website.
Otherwise, "Tuesday, 5pm will be the beginning of a complete civil disobedience campaign," it said.
Tamarod – Arabic for Rebellion – is a grassroots campaign which says it collected more than 22-million signatures seeking to withdraw confidence from the president.
The movement was behind Sunday's protests that saw millions of people take to the streets demanding his departure on the first anniversary of his inauguration as president. The turnout was described as the largest ever protest in the country's history.
Egypt has been deeply divided between Morsi's Islamist supporters who vowed to defend his legitimacy as the first Egyptian president elected in a free vote, while his opponents accuse him of installing an Islamist monopoly in state institutions.
Five people were killed when clashes broke out in the evening and another person died overnight from injuries, a ministry official told a private satellite channel.
Hundreds of demonstrators spent the night in Cairo's iconic protest site Tahrir Square and outside the presidential palace, after the army estimated that millions took to the streets to call for Morsi's resignation.
Some protesters had attacked the Cairo headquarters of the Muslim Brotherhood, from which Morsi hails.
Television pictures showed the building on fire as dozens of people attacked it, throwing stones and fire bombs. Supporters of the Brotherhood fired buckshot at the attackers in a bid to repel them, an AFP journalist at the scene said. Later, automatic weapons fire could be heard around the building.
'No half measures'
In its statement, Tamarod called on the army, the police and the judiciary to stand with the protesters.
It urged "state institutions including the army, the police and the judiciary to clearly side with the popular will as represented by the crowds" on Sunday.
It rejected presidential calls for dialogue, saying: "There is no way to accept any halfway measures.
"There is no alternative other than the peaceful end of power of the Muslim Brotherhood and its representative, Mohamed Morsi." – AFP