Polls opened in the capital Kigali on time at 7:00 am (0500 GMT), an AFP reporter said.
“Paul Kagame has brought peace,” said Eridigaride Niwemukobwa, 67, holding up her voter card proudly, but adding she did not know how long it may be possible for Kagame to run Rwanda if the constitutional changes pass.
“There is no secret, I will vote yes,” said Saidi Alfred, waiting in a line of around a hundred people to vote at a school in Kigali as polls opened. “It is because we want the president to continue to lead us.”
The amendment would allow Kagame to run for a third seven-year term in 2017, at the end of which the new rules come into force and he will be eligible to run for a further two five-year terms.
The expected changes have been denounced by Washington and Brussels as undermining democracy in the central African country.
Some 40,000 Rwandans abroad were eligible to vote on Thursday, but the main polls with some 6.4 million registered to vote are on Friday. Polls are scheduled to close at 3:00 pm (1200 GMT).
Kagame has run Rwanda since his ethnic Tutsi rebel army, the Rwanda Patriotic Front (RPF), ended a 1994 genocide by extremists from the Hutu majority, in which an estimated 800,000 people were massacred, the vast majority of them Tutsis.
The issue of long-serving rulers clinging to power has caused turmoil in Africa, where some heads of state have been at the helm for decades.
In response to criticism, Kagame has said that “other nations” should not interfere with the country’s internal affairs, or his people’s wishes.
Kigali’s pro-government New Times newspaper wrote in an editorial on Friday saying it expected the changes to pass.
“The referendum vote can only position him (Kagame) toward a path of choosing to continue stewardship of the country that has shaped from the ashes of the 1994 genocide against the Tutsi to the current glitter that it is,” the newspaper read.
Provisional results are expected late on Friday night, with final results to be announced before Monday, National Electoral Commission (NEC) executive secretary Charles Munyaneza has said.