The new charter aims to replace the constitution passed under former president Mohammed Morsi months before he was deposed by the army in July 2013 after nationwide protests against his Islamist rooted regime.
The two-day voting is also the first test for the army-backed coup that ousted Morsi and his Muslim Brotherhood, which has since been designated a terrorist group. The Brotherhood has called for a boycott of the polls.
Three people were killed in clashes with security forces in Sohag city while eight other deaths were reported.
Another 28 people were injured during the day.
Also Read
Earlier in the day, caretaker Prime Minister Hazem el-Beblawi called on Egyptians to perform their "national duty" by casting their vote.
A comfortable "yes" vote and a respectable turnout would bestow legitimacy on the cascade of events that led to Morsi's ouster.
A huge security operation is being mounted amid fears of violence. The Interior Ministry says 200,000 police officers, 150 central security units and 200 combat groups are being deployed around polling stations on both days of voting.
Morsi supporters have said they would stage massive demonstrations and have labelled the draft charter a "constitution of blood". In response, the government has warned it would deal harshly with anyone interfering with the referendum.
The new charter, drafted by a liberal-dominated committee appointed by the military-backed government, would ban political parties based on religion, give women equal rights and protect the status of minority Christians.
But it also gives the military special status by allowing it to select its own candidate for the job of defence minister for the next eight years and empowering it to bring civilians before military tribunals.
General El-Sissi has yet to say outright whether he plans to seek the nation's highest office, but his candidacy appears increasingly likely.