Polls have opened in Cairo, Alexandria and eight other provinces for the first round of voting which is scheduled to close at 7.00pm (local time).
The rest of the country votes on December 22.
The vote is being staggered over two rounds to ensure there will be enough judges to monitor polling stations amid a rift within the judiciary over the referendum process.
Rival camps in Cairo protested for weeks in the lead-up to the referendum in mass demonstrations that left eight people dead and hundreds more injured.
Egyptian president Mohamed Morsi's vote, cast in a polling station close to his presidential palace in Cairo, was broadcast on state television.
Mr Morsi's determined backing of the constitution triggered the power struggle with the opposition who accuse the Islamists of overreaching, a claim that is backed by judges.
Unofficial results from the first round of voting are expected within hours after booths close.
Mr Morsi has ordered Egypt's military to help police maintain security until a result is finalised.
Interior ministry officials told AFP that 130,000 police and 120,000 soldiers were being deployed.
The Islamist Muslim Brotherhood, which supports Mr Morsi, has organised large rallies and campaigned in favour of the draft constitution.
The National Salvation Front, the main opposition coalition, mulled a boycott before urging Egyptians to vote against the charter, which rights groups say limits the freedoms of minorities and women.
In a small queue at a Cairo school serving as a polling station, watched over by police and soldiers, several people said they were voting against the constitution.
"I'm voting because I hate the Muslim Brotherhood, it's very simple. They are liars," said Abbas Abdelaziz, a 57-year-old accountant.
Another man, 65-year-old Ali Mohammed Ali, said he regretted voting for Mr Morsi.
"I voted for Morsi and it was a mistake, a big mistake," he said.
"This constitution is bad, especially because it doesn't forbid child labour and opens the way for the marriage of minors."
Nagat Radi, a veiled woman in her 50s, said many articles in the draft constitution were problematic "and will hurt our country and our children".
"The people are going in one direction and the Brothers in another," she said.
"Those voting [in favour of the draft constitution] believe it is a gesture of piety and obedience to the president."
Others were in favour of the proposed charter.
Enayat Sayyed Mostafa, a retired woman, said the proposed constitution will help bring stabilisation to Egypt.
"I'm voting for stability and for Dr Morsi's promised program of renewal," she said.
"I have gone over the text to compare it with what the opposition is saying, and what they say is false. It's a good constitution."
The referendum was only made possible after Mr Morsi assumed sweeping powers that stripped courts of the right to annul the Islamist-dominated constituent assembly that drafted the charter.
He was forced to rescind his powers after mass protests outside his palace in northern Cairo led to the worst violence between the opposition and Islamists since his election in June.
International watchdogs, including the UN human rights chief, the United States and 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 the proven ability of the Muslim Brotherhood movement to muster voters was likely, but not certain, to ensure that the draft constitution is passed.