Next week's second round is likely to give another "yes" vote as it includes districts seen as more sympathetic towards Islamists, meaning the constitution would be approved. But a close win would give President Mohamed Mursi limited cause for celebration as it would demonstrate a wide rift in a country where he needs to build consensus on tough measures to fix a fragile economy.
The Muslim Brotherhood's party, which propelled Mr Mursi to office in a June election, said 56.5% backed the text. Official results are not expected until after the next round.
Mr Mursi and his backers say the constitution is vital to move Egypt's democratic transition forward. Opponents say the basic law is too Islamist and tramples on minority rights.
The build-up to Saturday's vote was marred by deadly protests after Mr Mursi awarded himself extra powers and then fast-tracked the constitution through an assembly dominated by his Islamist allies. However, the vote passed off calmly.