Petrol bombs were thrown and a number of people were injured, amid reports of shots being fired.
Supporters of President Mohamed Morsi dismantled tents set up outside the presidential palace by Morsi critics.
Vice-President Mahmoud Mekki has said a referendum on the draft will go ahead on 15 December despite the unrest.
But he indicated that changes could be made after the vote, saying the “door for dialogue” remained open.
He urged critics of the draft document to put their concerns in writing for future discussion.
Critics say the draft was rushed through parliament without proper consultation and that it does not do enough to protect political and religious freedoms and the rights of women.
The draft added to the anger generated by Mohamed Morsi passing a decree in late November which granted him wide-ranging new powers.
On Wednesday afternoon, supporters of Mohamed Morsi’s Muslim Brotherhood movement rallied outside the presidential palace, where the mainly secular opponents of the president were already staging a protest.
The pro-Morsi group chanted “The people want to cleanse the square” and “Morsi has legitimacy”, AFP news agency reported.
Stones and petrol bombs were thrown, before the Morsi supporters dismantled some of the tents set up by their opponents.
There were also reports of gunfire. Witnesses reported seeing a number of wounded people.
AFP said the anti-Morsi group had fled the area.
In a joint news conference, Mohamed ElBaradei, Amr Moussa and other leading opposition figures said they held Mohamed Morsi fully responsible for the violence.
Mohamed ElBaradei said they were “ready for dialogue, however we are ready to go to the streets”.
In a news conference broadcast earlier on state television, Mahmoud Mekki said there was “real political will to pass the current period and respond to the demands of the public”.
But he said there “must be consensus” on the constitution, and that “the door for dialogue is open for those who object to the draft”.
“I am completely confident that if not in the coming hours, in the next few days we will reach a breakthrough in the crisis and consensus,” he said.
He proposed that the opposition put their concerns about particular parts of the constitution into writing, but that this was “not a formal initiative but a personal idea”.
There are mixed messages coming from the government.
It has spoken about the need for dialogue for some time but has offered few concrete concessions which would end the crisis, he adds.
On Tuesday, tens of thousands of anti-Morsi demonstrators besieged the palace, clashing with police who fired tear gas.
Eighteen people were slightly injured in the brief burst of violence, the official Mena news agency reported.
At one point, the security forces issued a televised statement saying President Mohamed Morsi had left the building.
Many of those gathered outside the palace, in the suburb of Heliopolis, chanted slogans similar to those directed against the regime of former President Hosni Mubarak during the uprising in February 2011.
President Mohamed Morsi adopted sweeping new powers in a decree on November 22nd, and stripped the judiciary of any power to challenge his decisions.
Mohamed Morsi, who narrowly won Egypt’s first free presidential election in June, says he will give up his new powers once a new constitution is ratified.
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