It is said to be the most closely fought election since independence.
The election was delayed by six weeks to allow the army to recapture territory from militant Islamist group Boko Haram.
The two main presidential candidates have pledged to prevent violence during the election and its aftermath.
Despite reports of delays in some areas, election body spokesman Kayode Idowu told the AFP news agency that polling stations have opened and that “accreditation has started”.
Voters need to register first using biometric cards with their fingerprints before they can cast their vote later.
At some polling stations, card readers appear to be working slowly or not at all, BBC reporters on the ground say.
President Jonathan himself needed more than 20 minutes to register in his home village of Otuoke, although his spokesman says his accreditation is now complete.
The People’s Democratic Party (PDP) has dominated Nigerian politics since 1999, but the All Progressives Congress (APC) is viewed as a serious challenge.
Some 800 people were killed after the 2011 contest between Mr Jonathan and Gen Buhari, a former military ruler.
Voters in 36 states and the Federal Capital Territory of Abuja – the capital – will also elect members of the house of representatives and the senate.
On Friday, the Nigerian army said it had retaken the town of Gwoza, believed to be the headquarters of Boko Haram, one of the last places still under its control.