The 25-member US-appointed council had previously missed a 28 February deadline to strike a deal amid divisions over Islamic law, Kurdish autonomy demands and women in government.
One council spokesman confirmed: "Islam will be the official religion of the state and it will be a source ... of legislation."
"Federalism also has been approved as a form of government," he said, adding details of the agreement would be announced on Wednesday.
The constitution is to provide a framework for an interim government set to take power on 30 June from the US-led administration in Baghdad.
The council brings together leaders from Iraq's various ethnic and religious groups - comprising of Shia, Sunnis, Kurds and Turkmen.
During discussions on Friday, several Shia members of the council walked out, angered by the cancellation of a previous ruling that would have made divorce and inheritance subject to the rulings of religious law.
"Islam will be the official religion of the state and it will be a source ... of legislation. Federalism also has been approved as a form of government"
Iraqi Governing Council spokesman
Other sticking points had included defining the role of women in a future government, with some council members pushing for the constitution to set out a specific quota for the number of female representatives or parliamentarians.
There had also been furious debate over federalism, with the Kurds, who have effectively had self-rule in three northern provinces of Iraq since the 1991 Gulf war, pushing to enshrine that autonomy in the document.
Before Monday's announcement, an official in the US-led administration said occupation administrator Paul Bremer was unlikely to use his veto powers on a constitutional deal.
Officials said the negotiations were spirited and at times heated, but always respectful.