The BBC's Bethany Bell, outside Cairo university, says a crowd is chanting "raise your head high, your president is Morsi"
Islamist backers of Egypt's President Mohammed Morsi are holding mass rallies to support his sweeping new powers and the drafting of a constitution.
The demonstrations in Cairo come after days of rival protests by supporters and opponents of the president.
His opponents are angry that the draft constitution was hastily passed by an Islamist-dominated assembly on Friday.
The assembly acted before Egypt's top court could meet to rule on whether the body of MPs should be dissolved.
Senior judges have been in a stand-off with the president since he granted himself sweeping new powers last week.
Tens of thousands of supporters of Mr Morsi, carrying flags and portraits of the president, gathered outside Cairo university on Saturday.
"The people support the president's decision!" they chanted, while a banner read: "The people want the implementation of God's law."
Riot police stood by, with roadblocks erected to contain crowds.
Pro-Brotherhood crowds also demonstrated in other Egyptian cities. In Alexandria clashes broke out between supporters of Mr Morsi and unidentified activists, the state-run Mena news agency reported.
Mr Morsi's supporters in the Muslim Brotherhood and other Islamist parties had called for a huge turnout to show that the president's recent moves were supported by the public.
Under an emergency decree issued last week, Mr Morsi's decisions cannot be revoked by any authority, including the judiciary, until the new constitution has been ratified and a fresh parliamentary election held.
The decree also states that the courts cannot dissolve the constituent assembly.
Mr Morsi says he will give up his extraordinary powers once the new constitution is approved by a referendum.
Later on Saturday members of the constituent assembly are due to hand him the draft of the constitution they quickly adopted on Friday. The president is then expected to ratify the draft and schedule a referendum.
The BBC's Jon Leyne in Cairo says Mr Morsi may announce the referendum on Saturday and hold it in as little as two weeks' time.
Our correspondent says the key question will be whether the opposition can mobilise its support and get it to the ballot boxes to vote in the referendum.
On Friday, the president's opponents rallied in Cairo's Tahrir Square, chanting slogans including "The people want the fall of the regime!" - one of the rallying cries against former President Hosni Mubarak, who was toppled last year.
The extent of Mr Morsi's new powers has raised fears that he might become a new dictator.
UN Human Rights Commissioner Navi Pillay has written to the president, asking him to reconsider his decree.
In her letter, Ms Pillay "warned that approving a constitution in these circumstances could be a deeply divisive move", her spokesman said.
Mr Morsi's supporters point to the fact that he is Egypt's first freely elected president and argue that liberals and secularists do not represent the vast majority of Egyptians.
The presidential decree of 22 November gave the 100-member constituent assembly until January to complete the draft constitution.
When the Supreme Constitutional Court, Egypt's highest judicial authority, said it would soon rule on challenges to the process, supporters of the president in the assembly decided to rush through the draft.
During a marathon session that began on Thursday and continued through the night, the assembly voted on and passed all 234 articles.
Among the historic changes to Egypt's system of government, the draft limits the amount of time a president can serve to two four-year terms.
It also introduces some civilian oversight of the military establishment.
The draft keeps in place an article defining "principles of Sharia", or Islamic law, as the main source of legislation.