Mr Mursi, Egypt’s first freely-elected civilian president, is speaking on the eve of his swearing-in.Thousands had gathered in the square to protest against decrees issued by Egypt’s military rulers before Mr Mursi’s election.
Addressing the crowd, Mr Mursi promised to be a “president for all Egyptians”, adding: “You are the source of all authority and legitimacy.”
“I promise you that I will not give up on any of the powers given to the president,” Mr Mursi said, in a veiled reference to the Scaf’s recent decrees.
Mr Mursi is due to be sworn in on Saturday before the country’s Supreme Constitutional Court.
Spokesmen for the Islamist Muslim Brotherhood, to which Mr Mursi belongs, had previously said he would take his oath before parliament, which was dissolved by the Supreme Council of the Armed Forces (Scaf) last week.
The assembly, elected last November, was dominated by the Muslim Brotherhood-backed Freedom and Justice Party (FJP) and other Islamists.
The Scaf issued a declaration giving itself sweeping legislative powers and control over defence policy, and announcing the appointment of a panel to write a new constitution.
Another controversial decision by the Scaf to give military police powers of arrest was suspended by a court earlier this week.
Handling relations with the Scaf is likely to be a key test for Mr Mursi as he begins his term of office.Protesters gathered in Tahrir Square on Thursday chanted slogans against the military and in support of Mr Mursi.
“I’m here to tell the military council that we, the people, elected parliament so it is only us, the people, who can dissolve it,” Intissar al-Sakka, a protestor from the FJP, told the Reuters news agency.
The Scaf had previously said it would hand over power to Mr Mursi by the end of the month.
However, Scaf member Major-General Mohamed al-Assar told Egyptian media earlier this week that the head of Scaf, Field-Marshal Hussein Tantawi, would remain as defence minister under Mr Mursi.
Earlier, Mr Mursi performed Friday prayers at Cairo’s al-Azhar mosque, one of the most prominent seats of learning in Sunni Islam.
He has sought to allay fears among some secular and Coptic Christian Egyptians that he will use his presidency to impose Islamic law.
Mr Mursi’s campaign has said he plans to appoint a woman and a Coptic Christian