Brotherhood members, who earlier in the day gave a call for a huge rally near the presidential palace in support of President Mursi, boxed opposition protesters in from two sides, leading to scuffles, Al-Nahar TV channel reported.
The anti-Mursi protesters were staging a sit-in outside the palace when they were attacked.
Former presidential candidate Amr Moussa, who is also chairman of the Egyptian National Congress, warned that the clashes will further heat up the situation.
The Constitution Party also warned of "dire consequences" for what it described assaults on peaceful protesters in Tahrir Square and outside the palace.
"We are warning of dire consequences for mounting calls by figures affiliated with the Muslim Brotherhood and the Islamist current," the party said in a statement, which claimed that the president's supporters had called for massive protests "and even jihad, according to Al-Misry-al-Youm.
The clashes came after Mursi, who yesterday left the Palace through back gate, today returned to the complex.
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Meanwhile, Prosecutor General Talaat Ibrahim Abdallah yesterday ordered a probe against three former presidential candidates accusing them of espionage and conspiring a "Zionist plot" against the Islamist government.
Abdallah referred to the State Security Prosecution a complaint accusing Mohamed ElBaradei, Hamdein Sabahy and Moussa of spying and inciting the overthrow of the President.
Founder of the Constitution Party ElBaradei, Moussa and founder of the Popular Current Party Sabahy have declared their support to the ongoing anti-Mursi protests after the latter assumed absolute power through a decree last month, sparking the current crisis. MORE