This on Friday, the eve of a highly charged referendum on a new constitution, witnesses and state media said.
Police in riot gear quickly moved to restore order in the city, Egypt's second-biggest.
There were no immediate reports of injuries or arrests, but witnesses told AFP that the car of a cleric who had urged support of the referendum was set on fire.
Tensions are high over the staggered referendum, which is being held on this and the following Saturday, after weeks of protests and violence between rival camps in Cairo last week that killed eight people and injured hundreds.
Further mass rallies were taking place on Friday in Cairo by both sides. A pro-constitution rally by the Muslim Brotherhood backing Morsi gathered more than 2 000 people.
Egypt's mainly secular opposition has slammed the draft charter as divisive and a possible attempt by Islamists to introduce sharia-style law.
Mohamed ElBaradei, a former UN nuclear energy agency chief who heads the opposition National Salvation Front, said on his Twitter feed: "Insistence on referendum in an explosive, polarized, chaotic and lawless environment is leading country to the brink."
The Front on Friday was holding last-ditch rallies against the new constitution. It has said it could yet call a boycott if its "deep concern" over the referendum's fairness turns out to be founded.
Morsi has ordered Egypt's military to help police maintain security until the results of the referendum are known.
A total of 130 000 police and 120 000 soldiers will be deployed, interior ministry and military officials told AFP.
Polling in the referendum was staggered over a week because of a shortage of judges willing to provide the statutory supervision for the vote.
Voting will begin on Saturday in Cairo and Alexandria and in eight other provinces. A week later, on December 22, the other half of the country will vote.
A divided Eqypt
Morsi's Muslim Brotherhood and ultra-orthodox Salafist groups backing the draft charter have been campaigning for days for Egypt's 51 million voters to approve it.
A senior Islamist official involved in drafting the charter, Amr Darrag, branded opposition attacks against the document an "unjust campaign to paint a lying picture of the constitution."
The opposition took out half-page advertisements in major independent dailies describing the charter as "a constitution that divides Egypt."
The opposition – which initially wanted the referendum postponed – only started urging a 'no' vote on Thursday.
"It's you who will pay the price if you vote 'yes.' 'No' to the constitution," said an online campaign advertisement by the opposition April 6 youth movement.
International watchdogs, including the UN human rights chief, and the United States and the European Union, have expressed reservations about the draft constitution because of loopholes that could be used to weaken human rights, the rights of women and the independence of judges.
Passing of constitution 'will not silence opposition'
Analysts said the proven ability of the Muslim Brotherhood to muster voters was likely – but not certain — to ensure that the draft constitution is passed.
If it is adopted, "it will exacerbate political tension and result in more acrimony," Hani Sabra, an Egypt specialist for the Eurasia Group, said in an evaluation note.
"The Brotherhood and Morsi believe that if the constitution is adopted, it represents a mandate to pursue their policies.
However, majority support does not translate into the bureaucracy falling in line.
It will also not silence the opposition," he wrote. – Sapa-AFP