Supporters of Islamist Mr Mursi have been staging protests almost daily in Egypt since the army deposed him on July 3 in response to mass protests against him.
Yesterday's clashes in Cairo took place in the upmarket district of Mohandeseen, when a march by Muslim Brotherhood supporters came face-to-face with an opposing crowd.
The Mursi supporters were holding placards showing the four-finger logo of solidarity with those killed when security forces razed pro-Mursi protest camps in Cairo last August.
They chanted "down, down with military rule!"
Similar pro-Brotherhood protests were staged in other parts of Cairo along with the Suez Canal cities of Suez and Port Said. Most set off from mosques after yesterday's noon prayers.
The police arrested eight pro-Brotherhood protesters in Cairo for "inciting riots", according to the state run news agency Al-Ahram.
Two weeks ago a new law was promulgated that banned protests near or originating from places of worship, and made it compulsory to seek Interior Ministry permission for protests.
A ministry official said no such request had been filed for yesterday's protests.
About 180 Brotherhood protesters were arrested during similar protests on Friday of last week.