Mohammed Morsi has called for talks with the opposition in a bid to end the new political crisis engulfing Egypt.
The Egyptian President gave a national television address on Thursday evening in a bid to defuse the country's worst crisis in nearly two years.
He invited the opposition to a "comprehensive and productive" dialogue on Saturday at his presidential palace but gave no sign he would offer meaningful concessions.
Demonstrators responded by demanding the "downfall of the regime", using the chants that brought down previous leader Hosni Mubarak. Some protesters also raised their shoes in contempt.
Members of the main opposition coalition, the National Salvation Front, said they were assessing the offer but it was immediately rejected by the pro-democracy movement "April 6".
Fighting outside Mr Morsi's Cairo palace has already left at least six dead and almost 700 injured and the Republican Guard intervened on Thursday to stop the violence.
The crisis was sparked by Mr Morsi's decree on November 22, giving himself wide powers and protecting himself from judicial review.
The opposition has previously demanded that the president scrap his decree, postpone the referendum and redraft the constitution.
As well as drawing up a political roadmap, the leader said the talks would aim to resolve the fate of the upper house of parliament after the Islamist-dominated lower house was dissolved in June, the election law and other issues.
"I call for a full, productive dialogue with all figures and heads of parties, revolutionary youth and senior legal figures to meet this Saturday," he declared.
Several thousand opposition protesters near the palace waved their shoes in derision after his speech and shouted "Killer, killer" and "We won't go, he will go" - another of the slogans used against Mubarak in last year's revolt.
The Cairo headquarters of the Muslim Brotherhood, the group that propelled Mursi to victory in a June election, was set ablaze. Other offices of its political party were attacked.
This week's violence reflects the widening rifts in the most populous Arab nation, where contrasting visions of Islamists and their liberal rivals have complicated a struggle to embed democracy after Mubarak's 30 years of one-man rule.
Mr Morsi said he did not insist on keeping his actions shielded from legal challenge, adding that his entire decree would lapse after the constitutional referendum, regardless of its result.
He said a new constituent assembly would be formed to redraft the constitution if Egyptians rejected the one written in the past six months by an assembly dominated by Islamists.
The Republican Guard, an elite unit whose duties include protecting the presidential palace, had ordered rival demonstrators to leave by mid-afternoon.
Supporters of Mr Morsi withdrew, but opposition protesters remained, kept away by a barbed wire barricade guarded by tanks. By evening their numbers had swelled to several thousand.
The military played a big role in removing Mubarak duringlast year's popular revolt, taking over to manage a transitional period, but had stayed out of the latest crisis.
Thousands of supporters and opponents of Mr Morsi had fought well into Thursday's early hours, using rocks, petrol bombs and guns.
Before Mr Morsi's speech, opposition groups had called for protests after Friday prayers aimed at "the downfall of the militia regime", a dig at what they see as the Brotherhood's organised street muscle.
A communique from a leftist group urged protesters to gather at mosques and squares across Egypt, and to stage marches in Cairo and its sister city Giza, converging on the presidential palace.