Counting was under way early Sunday after a first-round referendum on a divisive new constitution pushed through by President Mohamed Morsi and his Islamist allies despite weeks of opposition protests.
Polling stations in half the country, including the biggest cities of Cairo and Alexandria, were tallying the results from Saturday's voting.
The second round of the referendum is to be held next Saturday, after which the official result is to be given.
Morsi's Muslim Brotherhood and main media outlets said that, based on unofficial figures, it appeared that the polling was trending around 60 percent support for the draft charter.
But the opposition insisted its preliminary figures suggested that 66 percent of the voters had rejected the proposed constitution. It claimed the Muslim Brotherhood had sought to "rig" the vote.
The official results will be announced after the second round on Saturday.
If those contradictory positions are maintained, Egypt's turmoil of the past three weeks over the draft constitution will not subside.
Violent clashes claimed eight lives on December 5 amid a highly polarised political climate.
Late Saturday, riot police fired tear gas to disperse dozens of hardline Islamists who attacked the central Cairo headquarters of the opposition liberal Wafd Party with fireworks and stones, officers at the scene told AFP.
On Friday, clashes between stone-throwing and sword-wielding Islamists and opposition supporters erupted in Egypt's second city of Alexandria, injuring 23 people according to the official MENA news agency.
Signup to our newsletter and follow us on Facebook and Twitter!
To ensure security, 120,000 troops were deployed to reinforce 130,000 police.
Voting was being staggered, with half the country casting their ballots on Saturday and the other half a week later.
The Muslim Brotherhood has thrown its formidable organisational machine behind a campaign in favour of the draft constitution.
The proposed charter "offers rights and stability," said one Cairo voter who backed it, Kassem Abdallah.
It will help Egypt "return to normal", agreed another, Ibrahim Mahmoud, a teacher.
But many opposition voters were especially hostile toward the Brotherhood, which the Front believes wants to usher in strict Islamic sharia laws.
Abbas Abdelaziz, a 57-year-old accountant, said he voted against the charter "because I hate the Muslim Brotherhood. It's very simple. They are liars."
Sally Rafid, a 28-year-old Christian, said: "There are many things in the constitution people don't agree on, and it's not just the articles on religion."
International watchdogs, the UN human rights chief, the United States and the European Union have expressed reservations about the draft because of loopholes that could be used to weaken human rights, including those of women, and the independence of the judiciary.
Analysts said it was likely -- but not certain -- that the draft constitution would be adopted.
Whatever the outcome, "lasting damage to the civility of Egyptian politics will be the main outcome of the current path Morsi has set Egypt on," one analyst, Issandr El Amrani, wrote for his think tank, the European Council on Foreign Relations.
"If the 'no' vote wins, the Morsi presidency will have been fully discredited and the pressure for his resignation will only increase," he said. "If 'yes' wins, the protest movement is unlikely to die down, (and) may radicalise."