With official results released from 14 of the Nigeria’s 36 states, Mr Abubakar is in second place, with around 33 per cent of the vote. Mr Obi is third, with 18 per cent.
But a group of angry protesters have gathered outside the national collation centre in the capital, Abuja.
They carried placards denouncing the electoral commission which has led to protests amid election result chaos.
Opposition parties in Nigeria have called for the presidential election to be scrapped, describing it as a sham.
The People’s Democratic Party (PDP) and the Labour Party (LP) said results had been manipulated, compromising the poll.
Governing party candidate, Bola Tinubu, is currently in the lead with just over 44 per cent of the vote, official figures released so far showed.
The Electoral Commission has not yet commented on the opposition’s demand.
The opposition parties were critical of the Independent National Electoral Commission (INEC) and its handling of the electronic voting system.
“I demand that this sham of an election be cancelled and we call on INEC to conduct fresh elections within the window period provided by the electoral act,” said Julius Abure, chairman of the Labour Party.
The African Democratic Congress (ADC) has also backed the call by Labour and the PDP, which is the main opposition party, for a fresh election to be held.
The PDP’s Atiku Abubakar and Labour’s Peter Obi were the main opposition presidential candidates in the elections held on Saturday.
“Everything happening there is all lies, all lies, lies… they are cooking up results,” one man told the BBC.
On Monday, Mr Abubakar’s and Mr Obi’s parties walked out of the venue where results were being announced.
They said there was 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 electoral commission 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 ran for the first time, promising to challenge the two-party system. —BBC
