As protesters prepared for a sit-in at the palace, police and military forces surrounded supporters of the Muslim Brotherhood, apparently to prevent them from getting near the palace, Ahram Online reported.
An al-Ahram website reported hundreds of Morsi opponents stormed a government office in Alexandria Friday in reaction to the killing of protesters Thursday by what they called "armed militias affiliated to the Muslim Brotherhood." Armed men chased the protesters out of the building, the report said.
Egyptian health officials said 11 people were injured in Friday protests in Cairo, and in Zagazig, where Morsi supporters and opponents clashed near his home. Officials said police had cordoned off the area around Morsi's home, denying access to thousands of protesters.
Mohamed ElBaradei, head of the opposition National Salvation Front, Friday called on Morsi to withdraw the controversial decree and put off the Dec. 15 referendum.
"I ask President Morsi in the name of the Egyptian conscience to heed these two demands tonight. I am betting on president Morsi's patriotism," ElBaradei said in televised speech.
Egyptian Vice President Mahmoud Mekki said Friday Morsi is willing to postpone a referendum on a new constitution but only on condition "the political forces who demand the delay of the referendum must provide guarantees that there will not be appeals [against the delay] in courts."
Egyptian protesters said Friday Morsi's call for dialogue gave no sign he is willing to give ground and that he has threatened his opponents.
Opponents say recent decisions by Morsi were "dictatorial."
The president condemned people involved in recent protests and spoke of "thuggery" and "terrorism."
Shaheer George, a spokesman for the liberal Free Egyptians, told Ahram Online a speech by the president was "a catastrophe that lacked a greater vision about the current crisis."
Ahmed Nour of the Revolutionary Socialists, who was among thousands of people outside the presidential palace, said Morsi's speech didn't include any suggestion the president is willing to have a real dialogue.
Nour said the president appears unwilling to talk about changing a draft constitution or reforming the constituent assembly.
"Instead, he threatened his opponents and incited his supporters against them."
Morsi has refused to step back from a controversial decree he issued, saying he respected peaceful opposition but won't tolerate with violence, CNN reported.
In a warning to "those who oppose me," Morsi, a member of the once-outlawed Muslim Brotherhood, condemned people involved in the protests and promised they would be held accountable.
"[They] will not escape punishment," he said Thursday.
Activists camped in Cairo's Tahrir Square chanted "Leave! Leave! Leave!" as Morsi talked.
Morsi recently decreed his decisions weren't subject to judicial review.
The opposition argues that a draft constitution, written by the country's Islamist-led Parliament, adversely affects civil liberties. It is scheduled to be put to a popular vote Dec. 15.
Morsi must roll back his edict and postpone the referendum on the constitution because it doesn't adequately represent or protect all Egyptians and was rushed through by Islamist lawmakers, opponents say.
Egyptian judges and media organizations have staged strikes to demonstrate disapproval.
Morsi has said his immunity from review would last until a new constitution is adopted.
Adel Saeed, a spokesman for Egypt's general prosecutor, said Friday ElBaradei, Hamdeen Sabahi and Amr Moussa were under investigation for allegedly "conspiring to topple" the government, CNN said.
All are well-known internationally -- ElBaradei is a Nobel Peace Prize laureate and Moussa is a former Arab League leader -- and are being investigated for their role in the opposition against Morsi, who defeated them in this year's presidential election.
The Muslim Brotherhood headquarters in Cairo and regional offices elsewhere were attacked and pro- and anti-government clashes persisted, authorities said.
Muslim Brotherhood spokesman Mahmoud Ghozlan said the headquarters of the group had been set on fire and ransacked Thursday, Ahram Online reported. Muslim Brotherhood officials also reported party offices in Ismailia, Suez and Monoufiya and elsewhere had been attacked or set on fire.
Eleven organizations said Morsi and the Muslim Brotherhood were behind the violence, several media outlets reported.
Morsi spoke of a "fifth column" of people who are clandestinely undermining the nation from within and said some protesters had been paid and armed to perpetrate "thuggery" and "terrorism."
"The time has come for them to be punished and penalized by the law. It is my duty to defend the homeland," Morsi said.
The White House said President Barack Obama called Morsi Thursday to express "deep concern" about the protesters' deaths and injuries and welcomed the Egyptian president's call for a dialogue, which Obama said "should occur without preconditions."