President Mohammed Morsi and his Muslim Brotherhood have endorsed the document, but critics say it is poorly drafted and favours Islamists.
Violent incidents were reported during voting, but not the widespread clashes that preceded the poll.
Early indications suggested that a majority of those who have voted so far may have backed the document.
A Muslim Brotherhood official told Reuters news agency that 56.5% backed the text, according to the party’s tally.
Opposition officials were also quoted as saying the “yes” vote appeared t
Continue reading the main storyOfficial results are not expected until after the second round, next Saturday.
Analysts say the second round is likely to give another “yes” vote as it includes districts seen as more sympathetic towards Islamists.
Turnout was unofficially estimated at around 32%. If confirmed, that would be far lower than the presidential or parliamentary elections following the fall of Hosni Mubarak.
A group of seven Egyptian human rights organisations reported widespread irregularities at polling stations.
They told a news conference they had reports of people falsely identifying themselves as judges, of women being prevented from voting, of some polling centres closing early and of some Christians being denied entry to polling stations.
The rights groups called on the election commission to prevent the same violations in the second round, and for the first round to be repeated.
The opposition National Salvation Front (NSF) reiterated its allegations that balloting had been “marred by irregularities and violations”.
It said it would “not recognise any unofficial result” and would wait for the formal tally after the second round.
The head of the NSF, Mohamed ElBaradei, who is a former chief of the UN nuclear energy agency, tweeted: “Country split, flagrant irregularities, low turnout, disillusion w(ith) Islamists on the rise. Illiteracy remains a hurdle.”
Saturday’s vote was held in Cairo, Alexandria and eight other provinces, a week before the rest of the country.
Polling reportedly had to be spread out because few judges were willing to supervise the referendum.
A constitution must be in place before elections can be held early next year.
Mr Morsi was elected president in June with just over half of the vote, more than a year after Hosni Mubarak was overthrown in a popular revolt in Egypt, the most populous country in the Arab world.
Opponents of the new constitution include liberals and others who want a more secular future for Egypt.
They say the charter would take away some of the new freedoms won in last year’s revolution.
The National Salvation Front had vehemently opposed the referendum, but this week said its supporters should go to the ballot boxes to vote “No”.
Supporters of the draft constitution have accused the opposition of sowing “lies and discord”.
During the first round of voting the headquarters of the opposition al-Wafd party in Cairo was attacked, reportedly by extremist Islamists.
Clashes were also reported in the cities of Dakahlia and Alexandria.
But BBC