Initial tallies suggest this week's two-day referendum passed with an overwhelming majority, in what the military-installed government said was a strong stamp of approval for the overthrow of Egypt's first freely elected president.
But Morsi's Islamist supporters, harried by a deadly crackdown since his removal in July, vowed more protests and faced off with police on Friday in clashes that killed four people.
The electoral commission is announce the official results of the referendum at a news conference scheduled for 1600 GMT.
Flagship state-owned newspaper Al-Ahram said Saturday the turnout was more than 40 percent of the country's 53 million registered voters, with the "overwhelming majority" voting yes.
It had earlier reported that 98 percent voted in support of the charter, an expected result after Morsi's Muslim Brotherhood movement and its Islamist allies boycotted the vote.
A Brotherhood-led coalition called for further protests culminating in rallies on January 25, the third anniversary of the 2011 uprising that overthrew veteran strongman Hosni Mubarak.
The new constitution will replace the one adopted under Morsi in a 2012 referendum and passed with 33 percent voter turnout. It has done away with much of the Islamist-inspired wording of Morsi's constitution.
On Friday, three men were killed in Cairo and another in clashes in Fayoum, southwest of the capital, the health ministry said, as police clamped down in what has become a weekly ritual in a massive crackdown on pro-Morsi protests.
The interior ministry said police arrested 123 suspected protesters in Cairo and other cities.
The government hoped a large turnout in the referendum held on Tuesday and Wednesday would bolster its democratic credentials and further marginalise the Islamists.
Army chief Abdel Fattah al-Sisi, the man who toppled Morsi, was closely monitoring the turnout as an indicator of support for a presidential bid later this year, military officials said.
'Sisi for presidency'
And the state-run Al-Akhbar newspaper threw its weight behind him on Saturday, declaring in a front-page banner that "All roads lead Sisi to the presidency of the republic".
The military-installed government has already said the vote showed support for Morsi's overthrow.
"This was also a referendum on June 30," said its spokesman Hany Salah, referring to the day when millions of protesters took to the streets demanding that Morsi resign.
The run-up to the vote was marred by arrests of activists who campaigned against the constitution.
"There was no real opportunity for those opposed to the government's roadmap or the proposed constitution to dissent," said monitoring group Democracy International, which observed the referendum.
The group said its monitors witnessed security forces and campaigning material inside polling stations, but there was "no evidence that such problems substantially affected the outcome of this referendum".
The government will still have to contend with near daily protests and a militant campaign in the Sinai that has killed scores of policemen and soldiers.
For the second time in a month, suspected militants in the restive peninsula blew up Friday a gas pipeline that feeds an industrial zone.
A coalition of Islamist opposition groups scorned the figures put out by state media and called for mass demonstrations on January 25 -- the third anniversary of Mubarak's overthrow.
"Let the putschists deceive themselves and hold fools' celebrations," the Anti-Coup Alliance said, claiming the referendum was a throwback to rigged elections under Mubarak.
The US administration is closely watching the results of Egypt's referendum, but has not yet decided whether to unfreeze some $1.5 billion (1.1 billion euros) in aid, the State Department said Thursday.
Washington suspended part of its aid -- the bulk of it military -- in October, angered that there had been little indication of a return to democracy after Morsi's overthrow.
The vote has put the Islamists on the back foot, and gives the government even less incentive to negotiate with the Muslim Brotherhood, which it has designated as a "terrorist" group.
Morsi himself has been in custody since his ouster and is currently standing trial in the first of three separate cases against him.