Police in the Egyptian capital, Cairo, have clashed with protesters angry at what they say is the rushed drafting of a new constitution and by President Morsi's recent extension of his powers.
Tens of thousands of people had gathered outside the presidential palace.
The police fired tear gas to disperse the crowd, some of whom managed to cut through barbed wire around the palace.
Mr Morsi was in the palace but left as the crowds grew, sources there said.
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.
Tear gas was fired after protesters managed to breach a barbed wire cordon surrounding the palace, correspondents say.
But the police quickly retreated, allowing protesters to get closer to the palace walls.
Eighteen people were injured in the brief burst of violence but none seriously, the official Mena news agency reported.
Large crowds remained outside as night fell, while thousands of demonstrators also gathered in Cairo's Tahrir Square.
"We won't be able to speak - there won't be a court that we can go talk to," one protester, Israa Wafid, told Reuters.
"He has made himself a fort and he says it is a temporary fort - this is something we cannot believe.
"We've spent 30 years being betrayed - we won't believe Morsi, he will remain seated in the chair and not leave it."
In a statement read out on television, Egyptian security forces called for calm among the protesters.
A sizeable crowd has also turned out in Egypt's second city Alexandria. Press action
Mohammed Morsi adopted sweeping new powers in a decree on 22 November, and stripped the judiciary of any power to challenge his decisions.
He has also called a nationwide referendum for 15 December on a new constitution, which opponents say has been rushed through and fails to protect the rights of minorities, particularly women.
Mr 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.
But his actions have brought out thousands, both his supporters and his opponents, in recent days.
Several newspapers refused to go to press on Tuesday, or printed blank front pages, in protest at what they say is the lack of press freedom in the constitution.