Morsi's aides and the Brotherhood, in turn, have accused the opposition of using the streets and condoning violence in an attempt to overturn the results of elections that Islamists have won repeatedly, securing their power.
They have tried to link the political opposition to a group of protesters called the Black Bloc, who wear black masks and have vowed to "defend the revolution." Officials and state media depict them as conspiratorial saboteurs, but the opposition says authorities are using the group as a scapegoat to justify a crackdown.
On Friday, thousands of residents marched through Port Said, located at the Suez Canal's Mediterranean end, pumping their fists and chanting, "Leave, leave, Morsi." They massed around the city's main security headquarters, but no clashes or violence was reported.
"The people want the Republic of Port Said," protesters chanted, voicing a wide sentiment among residents that they are fed up of negligence and mistreatment by central government and that they want to virtual independence.
Egypt's main opposition political grouping, the National Salvation Front, called for Friday's protests, demanding Morsi form a national unity government and amend the constitution. They say the unrest reflects the widespread discontent over Brotherhood attempts to rule alone and keep decision-making in its own hands.
"The policies of the president and the Muslim Brotherhood are pushing the country to the brink," the opposition said in a statement.
But there were signs of splits and confusion in the opposition ranks after leaders of the Front met for the first time with the Brotherhood as part of a dialogue hosted by Egypt's premier Islamic institution, Al-Azhar. The Front had previously refused talks with the Brotherhood until its conditions were met.
With Front leader Mohamed ElBaradei and the deputy leader of the Brotherhood at the same table, the gathering of a spectrum of politicians signed a joint statement denouncing violence.
The statement, known as the Al-Azhar Document, angered some in the anti-Morsi camp. It seemed to focus on violence by protesters with no mention of excessive force by police or the wider political issues.
"Al-Azhar's initiative talks too broadly about violence," a group of 70 activists, liberal politicians, actors and writers said in a statement criticizing the meeting. They said the document gives "political cover to expand the repression, detention, killing and torture by the hands of police for the authority's benefit."
The document "didn't offer solutions but came to give more legitimacy to the existing authority," it added.
Those who attended the Thursday's meeting defended the anti-violence initiative.
"We toppled down Mubarak regime with a peaceful revolution. We insist on achieving the goals the same way whatever the sacrifices and the barbaric suppression tactics," ElBaradei wrote in a tweet.
"No one can say no to an initiative to stop violence," said Ahmed Said, an opposition party leader.