The Interior Ministry said Wednesday 95 percent of participants approved the new governing document, Iran's Press TV reported.
"Turnout so far may exceed 55 percent and the approval of the constitution is perhaps more than 95 percent," Maj. Gen. Abdel Fattah Othman, the ministry's spokesman, told private satellite channel al-Hayat.
While earlier reports indicated final results would be available Thursday, Press TV said the official results are not expected until Sunday.
At least 11 people were killed and about 450 people, most of them members of the Muslim Brotherhood, were arrested after clashes broke out during the two days of voting, the Iranian network said.
Ahram Online had reported earlier Interior Minister Mohamed Ibrahim warned against vote tampering.
"Anyone who contemplates meddling with the referendum ballot boxes once they are closed will be shot with live ammunition," Ibrahim said Tuesday after violence during the first day of voting left at least nine people dead.
The vote was conducted amid political tensions and Egyptian security sources told Ahram Online the army and police secured polling stations across the country to prevent further unrest after Monday's violence.
A bomb exploded early Tuesday in a Giza neighborhood. No injuries were reported. Later in the day, at least nine people were killed in clashes in Sohag, Giza and the Beni Suef governorate between supporters of ousted President Mohamed Morsi and security forces, Ahram Online said.