Critics have said the proposals, if approved, would be a step backwards for democracy in Azerbaijan.
It would "place Azerbaijan among the backward countries," Isa Qambar, head of Musavat, an opposition party, said.
But supporters believe the change to the constitution will benefit the former Soviet state.
"I voted so that Ilham Aliyev may continue to be our president because he has brought us stability," Khatima Jabrailova, a 72-year-old pensioner, said.
Opposition groups have accused authorities of fixing the vote, harassing anti-government campaigners and controlling the media.
But despite these objections, the referendum has drawn little criticism from the United States or European Union.
Anti-government groups in Azerbaijan have accused the West of turning a blind eye to the actions of their government, in order to secure access to the country's vast oil and gas reserves in the Caspian Sea.