The referendum has deeply polarised the country and riled a secular-leaning opposition.
The Muslim Brotherhood, from which President Mohamed Morsi hails, and the official Al-Ahram newspaper reported that about 64% of votes cast were in favour of the new charter, after preliminary results were tallied from the second round vote on Saturday.
Turnout over both rounds was around 32%, according to the Muslim Brotherhood. The early results are based on reports from returning officials from the vast majority of stations over the two rounds, which were held a week apart.
The election committee will announce the final results within two days. "The Egyptian people continue their march towards finalising the construction of a democratic modern state, after turning the page on oppression," the Brotherhood's political arm, the Freedom and Justice Party, said in a statement.
In Washington, Republican Representative Ileana Ros-Lehtinen, who chairs the House Foreign Affairs Committee, called the vote "a defeat for the Egyptian people."
"We cannot celebrate the trade of an authoritative regime for an Islamic dictatorship," she added in a statement issued ahead of the announcement of any results.
On the eve of Saturday's polling, clashes in Egypt's second city Alexandria injured 62 people as stone-throwing mobs torched vehicles, underlining the deepening fractures in the Arab world's most populous nation.
Eight people were killed and hundreds more injured in clashes between rival demonstrators on December 5 outside Morsi's presidential palace in Cairo, in scenes not witnessed since the uprising that overthrew president Hosni Mubarak in early 2011.
Some 250 000 police and soldiers were deployed to provide security during the referendum.
The army has also positioned tanks around the presidential palace since early this month. The main opposition group, the National Salvation Front, alleged in a statement that ballot fraud had taken place, citing reports of laymen posing as judges supposed to oversee the vote.
Morsi's vice president, Mahmud Mekki, whose post is not mentioned in the new charter, announced on Saturday that he was resigning.
"Political work does not suit my professional character," he said in a statement, referring to his past as a respected judge.
State television reported that Central Bank chief Faruq El-Okda had also resigned, but later cited a cabinet source as denying this had happened.
A modest margin and a low turnout in the referendum is expected to embolden the opposition, which looks likely to continue its campaign against Morsi and his Muslim Brotherhood movement.
Once the constitution is ratified, the Islamist dominated senate will assume full legislative powers until a new parliament is elected in two months to replace the Islamist-dominated assembly a court annulled before Morsi's election in June.
On Saturday, Morsi appointed 90 additional senators, including eight women and 12 Christians, to further "national dialogue," his spokesman said in a statement.
Morsi rushed through the referendum after granting himself extensive powers in November that placed his decisions beyond judicial review, sparking the worst political crisis since his election.
At one polling station in Giza, southwest Cairo, 50-year-old housewife Zarifa Abdul Aziz said: "I will vote 'no' a thousand times. … I am not comfortable with the Brotherhood and all that it is doing."
However, 19-year-old law student Ahmed Mohammed said he voted "yes" because "Egypt needs a constitution to be stable."
Rights groups say the charter limits the freedoms of religious minorities and women, while allowing the military, which retains considerable influence over politics, to try civilians it believes "harm" the army.
Morsi had to split the voting over two successive Saturdays after more than half of Egypt's judges said they would not provide the statutory supervision of polling stations.
In Washington, Republican Representative Ileana Ros-Lehtinen, who chairs the House Foreign Affairs Committee, was in no doubt that the Islamists would claim victory.
"However, the stark reality is that this is a defeat for the Egyptian people," she said in a statement. "We cannot celebrate the trade of an authoritative regime for an Islamic dictatorship," she added.
Morsi's constitution lacks key democratic principles that allows for the oppression of secular and Christian Egyptians and fails to protect basic human rights.
The US must hold the Morsi government accountable for its actions and the Obama Administration must not simply grant Egypt funds whenever it asks.
We must use our aid as leverage to promote democratic reforms, support freedom of religion, and enshrine the protection of minority communities." – Sapa-AFP