There have been delays and technical problems, partly due to new biometric voter cards.
Eyewitnesses say 13 people have been killed in five attacks by gunmen, but it is not clear whether they are Boko Haram militants or political thugs.
The election is said to be the most closely fought since independence.
It was postponed from mid-February to allow the army to recapture territory from the Islamist militants of Boko Haram.
The two main candidates have pledged to prevent violence.
Voters are also electing members of the house of representatives and the senate.
By 12:30 local time (11:30 GMT), 81% of polling stations were open, according to the Transitional Monitoring Group (TMG), the largest body observing the elections.
Voters need to register first using biometric cards with their fingerprints before they can cast their vote.
However, at some polling stations, card readers are working slowly or not at all.
President Jonathan tried for some 50 minutes to register in his home village of Otuoke. A BBC reporter at the scene says he had to come back a second time and when the electronic registration failed again, he had to be accredited manually.
Problems were also reported from the north’s biggest city of Kano, where thousands of voters waited for election officials and voting materials to arrive.
“We’ve been here since six o’clock and now it’s half-past nine,” Ismail Omar, a 65-year-old builder told AFP news agency.
“No-one has shown up from Inec (the Independent National Electoral Commission)… This is a deliberate attempt to sabotage the elections.”
But Gen Buhari registered in his hometown Daura without any problems and he praised the accreditation system.
“If people are allowed to vote, rigging will actually be impossible under this system,” he said.
Voting has now begun in some areas.
The reported attacks have occurred in the north-eastern Gombe state. Boko Haram had threatened to disrupt the poll.
As millions of voters queued outside polling stations, the Inec website was briefly hacked.
A group calling itself the Nigeria Cyber Army placed a message on the website warning Inec not to rig the elections, before normal service was resumed.
Other incidents reported include a controlled car bomb explosion at polling station in Enugu state after the authorities discovered a car bomb and, according to the TMG, intimidation of election monitors at three polling stations in the south.
The People’s Democratic Party (PDP) has dominated Nigerian politics since 1999, but Gen Buhari’s 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, who alleged fraud.
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.