* 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
CAIRO, Dec 6 (Reuters) - Egypt's Islamist president, Mohamed Mursi, called on Thursday for a national dialogue after deadly clashes around his palace, where demonstrators responded by demanding the "downfall of the regime", using the chants that brought down Hosni Mubarak.
Mursi said in a televised speech that plans for a referendum on a new constitution on Dec. 15 were on track, proposing a meeting on Saturday with political leaders, "revolutionary youth" and legal figures to discuss the way forward after that.
The Republican Guard intervened on Thursday to halt violence outside the palace, where seven people were killed overnight in clashes between supporters and opponents of Mursi.
Members of the main opposition coalition, the National Salvation Front, said they were assessing the offer of talks to end a crisis sparked by Mursi's Nov. 22 decree awarding himself wide powers and protecting his decisions from judicial review.
The opposition has previously demanded that Mursi scrap his decree, postpone the referendum and redraft the constitution.
As well as drawing up a political roadmap, Mursi 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," Mursi declared, adding that the meeting would be at his official palace.
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.
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 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. 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.
The military played a big role in removing Mubarak during last 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 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.
Prosecutors investigating the unrest said Brotherhood members had detained 49 wounded protesters and were refusing to release them to the authorities, the state news agency said. The Brotherhood denied this, saying all "thugs" detained by its members had been handed over to police or the Republican Guard.
Before Mursi'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. "Egyptian blood is a red line," the communique said.
Hardline Islamist Salafis summoned their supporters to protest against what they consider biased coverage of the crisis by some private Egyptian satellite television channels.
Outside Cairo, supporters and opponents of Mursi clashed in his home town of Zagazig in the Nile Delta, state TV reported.
Egypt plunged into renewed turmoil after Mursi issued his Nov. 22 decree and an Islamist-dominated assembly hastily approved a new constitution to go to next week's referendum.
Since then six of the president's advisers have resigned. Essam al-Amir, the director of state television, quit on Thursday, as did a Christian official working at the presidency.
The Brotherhood's supreme guide, Mohamed Badie, called for unity, saying divisions "only serve the nation's enemies".
Mursi's opponents accuse him of seeking to create a new "dictatorship". The president says his actions were necessary to prevent courts still full of judges appointed by Mubarak from derailing a constitution vital for Egypt's political transition.
U.N. Human Rights Commissioner Navi Pillay urged the Egyptian authorities to protect peaceful protesters and prosecute anyone inciting violence, including politicians.
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, after previously firming on hopes that a $4.8 billion IMF loan would stabilise the economy. The stock market fell 4.6 percent.