* Mursi says constitutional referendum on track for Dec. 15
* President calls for national dialogue
* Opposition says assessing Islamist leader's offer
* Republican Guard restores order at presidential palace
Demonstrators rejected a call from Egypt's Islamist President Mohamed Mursi for a national dialogue after deadly clashes around his palace, demanding the "downfall of the regime" - the chant that brought down Hosni Mubarak.
Mursi said in a televised speech late on Thursday that plans were on track for a referendum on a new constitution on Dec. 15 despite clashes that killed seven people. He proposed a meeting on Saturday with political leaders, "revolutionary youth" and legal figures to discuss the way forward after that.
But a leading activist group rejected the offer, and fresh demonstrations were called for Friday.
The "April 6" movement, which played a prominent role in igniting the revolt against Mubarak said on its Facebook page that Friday's protests would deliver a "red card" to Mursi.
Egypt has been plunged into turmoil since Mursi issued a decree on Nov. 22 awarding himself wide powers and shielding his decisions from judicial review.
His Islamist supporters say the decree was necessary to prevent Mubarak-era judges from interfering with reforms. A constitution drawn up by a body dominated by Islamists is due to be put to a referendum next week.
The opposition has demanded that Mursi scrap his decree, postpone the referendum and redraft the constitution.
In his address, Mursi said: "I call for a full, productive dialogue with all figures and heads of parties, revolutionary youth and senior legal figures to meet this Saturday."
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.
The United States, worried about the stability of an Arab partner which has a peace deal with Israel and which receives $1.3 billion a year in U.S. military aid, had urged dialogue.
Mursi said 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.
The Republican Guard, an elite unit whose duties include protecting the presidential palace, restored peace on Thursday after a night of violence outside the palace, ordering rival demonstrators to leave by mid-afternoon.
Mursi supporters withdrew, but opposition protesters remained, kept away by a barbed wire barricade guarded by tanks. By evening their numbers had swelled to several thousand.
Thousands of supporters and opponents of Mursi had fought well into Thursday's early hours, using rocks, petrol bombs and guns. Officials said 350 were wounded in the violence. Six of the dead were Mursi supporters, the Muslim Brotherhood said.
Opposition groups have 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. "Egyptian blood is a red line," the communique said.
Hardline Islamist Salafis also summoned their supporters to protest against what they consider biased coverage of the crisis by some private Egyptian satellite television channels.
Since Mursi issued his decree, six of his advisers have resigned. Essam al-Amir, the director of state television, quit on Thursday, as did a Christian official at the presidency.
The Brotherhood's supreme guide, Mohamed Badie, called for unity, saying divisions "only serve the nation's enemies".
The Islamists, who have won presidential and parliamentary elections since Mubarak was overthrown, are confident they can win the referendum and the parliamentary election to follow.
As well as relying on his Brotherhood power base, Mursi may also tap into a popular yearning for stability and economic revival after almost two years of political turmoil.
Egypt's pound hit an eight-year low on Thursday, reversing gains made on hopes that a $4.8 billion IMF loan would stabilise the economy. The stock market fell 4.6 percent.