Nigerian APC’s candidate Bola Ahmed Tinubu has taken the lead ahead of his contenders Mr Obi and Mr Atiku after the coalition of 14 states of the total 36 states votes.
Mr Tinubu has won just over 44% of the tallied votes. His main rival Atiku Abubakar has almost 33% of the vote.
Labour’s Peter Obi has close to 18%, having caused an upset by defeating Mr Tinubu in the biggest city, Lagos.
On Monday, Mr Abubakar’s and Mr Obi’s parties walked out of the venue where results are being announced.
The Peoples Democratic Party (PDP) and the Labour Party have claimed there is a lack of transparency with the new electronic voter system.
This was the first national election where an electronic device had been used to accredit voters.
The Independent National Electoral Commission (Inec) has denied the opposition parties’ complaints.
Its chairman Mahmood Yakubu said the announcement of results would continue.
European Union observers said the electoral body’s poor planning and communication undermined trust in the process.
The PDP representative at the election centre in the capital, Abuja, described the process as fraudulent, and accused the governing All Progressives Congress (APC) of colluding with Inec.
The Labour Party asked for the announcements to be suspended, or for the election to be cancelled and rerun.
The APC said those dissatisfied with the results should go to court, and that the parties should first let the process run its course.
The APC and PDP have dominated Nigeria since the end of military rule in 1999.
Mr Obi – from the previously little-known Labour Party – ran for the first time, promising to challenge the two-party system.
He has the support of many young people, who make up a third of registered voters. There are 15 other candidates.
From the results announced so far Mr Tinubu has won the most votes in six states, Mr Abubakar five and Mr Obi three.
A candidate needs to have the most votes and a quarter of ballots cast in 25 of the 36 states plus Abuja to be declared the winner.
If those thresholds are not reached then there will a second round run-off between the top two candidates.
Credit: Nduka Orjinmo BBC News, Abuja