Cairo and other cities have suffered a series of often violent demonstrations over the past three weeks since Mr Mursi assumed sweeping new powers to push through the constitution, which he sees as a vital element of Egypt's transition to democracy after the overthrow of autocratic predecessor Hosni Mubarak last year.
At least eight people have died and hundreds have been injured, and a leading opposition figure has warned of more blood on the streets during the voting today and next Saturday on a draft that the opposition says is too heavily influenced by Islamists.
The referendum, held on two days because there are not enough judges willing to monitor all polling stations, asks Egyptians to accept or reject a basic law that must be in place before national elections can be held early next year – an event many hope can steer the Arab world's most populous nation towards stability.
To bolster support for the constitution, Islamists who propelled Mr Mursi to power in June's presidential election assembled at a mosque near the President's palace in Cairo.
They chanted: "We've come here to say 'yes' to the constitution. Long live President Mursi."
Members of the liberal, secular and Christian opposition also gathered outside the presidential palace, to protest against the basic law.
The first round of voting today will take place in Cairo and other major cities. Official results won't be announced until after the second round.