The Islamic political party said late Saturday results showed the charter was garnering 70 percent of the vote with more than 60 percent of the polling places counted, The New York Times reported.
The newspaper said Egyptian Vice President Mahmoud Mekki resigned Saturday. His position would be eliminated under the new constitution.
The former judge had originally submitted his resignation in early November, but stayed on during the turmoil leading up to the constitutional referendum.
"The nature of political work does not suit my nature as a judge," he said.
Once the votes are officially tabulated in the coming days, President Mohamed Morsi will deliver a nationally televised address urging unity and reconciliation, his aides said.
His advisers said he plans to appoint some of his opponents to the Islamist-dominated upper house of Parliament that will act as the main legislative body until upcoming elections for the lower house.
Ahram Online reported Morsi spokesman Yasser Ali announced the names Saturday of 90 members of the 270-member upper house, the Shura Council.
Ali said in a post on his official Facebook page, the appointments represent 17 political parties, of which 12 parties were not previously represented. The members also include eight women, 12 Coptic Christians, five Azhar representatives and two representatives of the January 25 revolution's injured.
Millions of Egyptians voted Saturday, but overall the turnout, like in the previous week's voting, was low, the Times said.
Salafist Nour Party and Brotherhood FJP Party members were at some polling stations throughout Egypt, campaigning on behalf of the draft of the constitution, Ahram Online reported.
"Voting 'Yes' will ensure Egypt enters a new era under an umbrella of justice and freedom that hasn't been witnessed in decades," said the Muslim Brotherhood's Supreme Guide Mohamed Badie.
Opponents of President Mohammed Morsi have been protesting the draft, the BBC reported.
"We know that if this constitution is passed, there will be no stability," opposition figure Mohammed ElBaradei said Thursday.
The voting came a day after Islamists clashed with secular rivals in Alexandria, hurling stones at each other outside a mosque. Police dispersed the crowd using tear gas; 32 people were injured in the incident.