Keita says the results were manipulated
But the candidate in third place, former Prime Minister Ibrahim Boubacar Keita, who was backed by Mali's Islamic leaders, has alleged fraud and vote-rigging.
The Independent Electoral Commission has also criticised the way the count has been handled.
Mr Keita's RPM party and five others allege that results were manipulated before being released by the interior ministry.
No conflict
Mr Cisse's campaign manager is the wife of the interior minister, Ousmane Sy.
Mr Sy has denied any conflict of interests.
The RPM is threatening to take the results to court but has asked their activists to remain calm.
"It's clear the vote has been manipulated, because in some zones, the number of votes cast was higher than the number of registered voters," Mr Keita told journalists.
"I am not a rabble-rouser... But if you rob and cheat your people, it is sometimes hard to keep them under control," he warned.
The RPM has called for a mass rally of its supporters in Bamako on Saturday.
'Unfair'
The president of Mali's Independent Electoral Commission, Moustaphe Cisse, told journalists that polling stations had not provided official electoral results, as required by the law.
It was also "unfair" that results from Bamako had not been released, he said.
The BBC's Joan Baxter in Bamako says that privately many observers have "serious reservations" about the counting of the votes.
Current President Alpha Oumar Konare is not standing, as he has served the constitutional limit of two terms.
Mali is often referred to by westerners as a model democracy in Africa, since multi-party elections were introduced by Mr Toure in 1992.
But our correspondent says many Malians are concerned about rising poverty and what they see as the spread of corruption.
Password
International election observers have agreed not to make an official statement until final results are declared.
They were initially expected on Tuesday but were delayed after a computer technician had a car accident.
Election officials said he was the only person with the password to access the election centre's computers - forcing the vote count to be suspended.
But our correspondent says that many people suspect officials used the accident as an excuse to suspend counting.