Voters turned out up in big numbers in Cairo and in most of the 10 provinces where a first round of voting was held on Saturday.
The state-run al-Akhbar newspaper reported that 59% of the votes counted so far were in favour of the Islamist-backed constitution.
The Muslim Brotherhood's Freedom and Justice Party put the figure at 57%.
President Mohammed Morsi's Brotherhood and its Islamist allies have campaigned heavily for the draft constitution, arguing that it is necessary to fast-track the transition from Hosni Mubarak's authoritarian rule.
The opposition, which has complained of irregularities during the vote, says the charter could undermine political rights and sideline minorities.
With voters queuing for hours in front of polling stations, monitors said the polling was marred by a number of violations, including mass voting, lack of full judicial oversight and illegal canvassing.
The official result is to be announced after the second and final round of voting takes place in the country's remaining 17 provinces on December 22. – DPA