The clashes erupted after the Muslim Brotherhood, the group that helped Mr Mursi win a presidential election in June, told its supporters to go to the palace where opponents had gathered to protest against the president's expanded powers.
Brotherhood supporters said they were shot at during the clashes, where rival sides brawled and threw stones at each other. One protester showed blood pouring from what he said was a gunshot wound to his leg.
Egyptian opposition leader Amr Moussa said Mr Mursi should make a formal offer for dialogue to end a crisis over the constitution, after the vice-president set out what he described as personal ideas to resolve the row.
Opposition politicians met to respond to the proposals laid out by Vice-President Mahmoud Mekky, who presented his ideas to reporters at the presidential palace. Outside the compound, Mr Mursi's opponents and supporters scuffled with each other.
"We are ready when there is something formal, something expressed in definite terms; we will not ignore it, especially if there is something useful," Mr Moussa said, adding he was holding talks with other politicians.
He said the terms used by Mr Mekky to describe his ideas meant they could not be con-sidered a formal or official offer with presidential backing.
Mr Moussa is a member of an opposition alliance, the National Salvation Front, which includes several prominent figures such as former UN nuclear agency chief and Nobel peace laureate Mohamed ElBaradei.
Another member of the alliance, Abdel Ghaffar Shokr, head of the Popular Socialist Coalition Party, said the opposition would issue a statement to respond to Mr Mekky. "Until now, we still hold our position that the decree should be cancelled and the constitution should be delayed," he said, referring to Mr Mursi's decree on November 22 that gave him extraordinary powers.
Mr Mekky said amendments to disputed articles could be agreed with the opposition.
A written agreement could then be submitted to the next parliament, to be elected after a referendum on the constitution on December 15.
On those ideas, Mr Shokr said it would be unusual for a con-stitution to be amended after just two months. A senior Muslim Brotherhood official said the ideas needed to be "crystallised".
Mr Mursi said he had expanded his powers to prevent courts still full of judges appointed by ousted strongman Hosni Mubarak from derailing a con-stitution meant to complete a political transition in Egypt, long a strategic ally of Washington and signatory to a 1979 peace deal with Israel.
During the clashes between rival groups outside the presidential palace, Islamist supporters of Mr Mursi tore down tents erected by leftist opponents, who had begun a sit-in.
"They hit us and destroyed our tents. Are you happy, Mursi? Aren't we Egyptians too?" asked one protester, Haitham Ahmed.
Opposition supporters streamed away from the palace as hundreds of Islamists arrived, shouting "God is greatest. The people support the president's decision."
Facing the gravest crisis of his six-month-old tenure, Mr Mursi has shown no sign of buckling, confident Islamists can win the referendum and a parliamentary election to follow.
US Secretary of State Hillary Clinton said dialogue was urgently needed on the new constitution, which should "respect the rights of all citizens".