outside a large media compound.
The clashes erupted as US Under Secretary of State William Burns landed in Cairo, the latest international envoy on a mission to press Morsi's backers and the interim government to resolve the standoff peacefully.
Meanwhile, Ayman al-Zawahiri, the Egyptian-born head of the militant Al-Qaeda organisation, accused the United States of "plotting" Morsi's overthrow with the Egyptian military and the country's Christian minority.
"Crusaders and secularists and the Americanised army have converged with Gulf money and American plotting to topple Mohamed Morsi's government," he said in a 15-minute audio recording posted on militant Islamist forums.
In his first comments since the July 3 coup, Zawahiri also attacked Morsi's secular opposition and Coptic Christians, who he said wanted a secessionist state in Egypt, and called for a mass movement to install Islamic law.
In Cairo, Morsi supporters began to march after Friday prayers, pouring out of several mosques in the capital.
An early evening protest outside Cairo's media production city descended into mayhem, with at least one protester wounded by birdshot.
Police fired tear gas at protesters who had set up tents and brick fortifications outside the compound. The protesters responded with stones.
A security source said protesters had "tried to storm" the compound, but the demonstrators said police fired tear gas when they tried to camp outside the compound.
"I am a Muslim, not a terrorist," they chanted.
The interior ministry accused the protesters of firing birdshot, wounding a conscript, and said police made 31 arrests.
Witnesses also reported clashes between residents in the Alf Maskan area and Morsi loyalists after they tried to set up a protest site.
Morsi supporters announced Friday evening marches to several security facilities, including the Republican Guard headquarters, where more than 50 demonstrators were killed last month.
While large crowds turned out to one march at the military intelligence headquarters, stopping short of the building and turning back after a brief protest, attempts to march to the Republican Guards appeared to have been called off.
Military vehicles were stationed on at least two side streets from the key Rabaa al-Adawiya protest site, and on the way from the site to the Republican Guard facility.
At a march on the state security police forces, protester Ward Abdel Moghny said demonstrators "went to them... to say we no longer fear you."
Morsi's supporters have remained defiant in the face of mounting threats from the army-installed interim government.
The interior ministry has urged those dug in at protest sites in Rabaa al-Adawiya and Nahda squares "to let reason and the national interest prevail, and to quickly leave".
State-owned Al-Ahram newspaper reported Friday that police had a plan to disperse the sit-ins but were holding out for a peaceful resolution.
Vice President Mohamed ElBaradei called for a halt to violence in an interview with the Washington Post.
"Once we do that, we immediately have to go into a dialogue to ensure that the Brotherhood understand that Mr. Morsi failed. But that doesn't mean that the Brotherhood should be excluded in any way."
More than 250 people have been killed since Morsi's ouster and diplomatic efforts to avoid further bloodshed have picked up pace.
US envoy Burns arrived in Cairo Friday night on the heels of trips by the European Union's Middle East envoy Bernardino Leon and German Foreign Minister Guido Westerwelle.
A senior member of the Freedom and Justice Party, the Brotherhood's political arm, said the European envoys had asked them to end their sit-ins.
Morsi supporters have also been angered by comments from US Secretary of State John Kerry, who told Pakistani television that Egypt's military was "restoring democracy" when it ousted the Islamist leader after protests against him.
"Is it the job of the army to restore democracy?" asked Muslim Brotherhood spokesman Gehad al-Haddad in a statement on Friday.
Rights group Amnesty International meanwhile demanded an investigation into allegations that Morsi's supporters tortured opponents in Cairo near their protest camps.
It said opponents of the Islamist leader reported being "captured, beaten, subjected to electric shocks or stabbed by individuals loyal to the former President."
Morsi has been formally remanded in custody on suspicion of offences when he broke out of prison during the 2011 revolt that toppled former president Hosni Mubarak.
Prosecutors have also referred three top Muslim Brotherhood leaders, including supreme guide Mohamed Badie, for prosecution on allegations of inciting the deaths of demonstrators.
Morsi was detained hours after the coup and is being held at an undisclosed location, where his family has been unable to see him.
EU foreign policy chief Catherine Ashton met Morsi on Tuesday and said he was "well".