Buhari's campaign spokesman said incumbent President Goodluck Jonathan had called the retired general to concede defeat in the most closely fought election in Nigeria's history.
If confirmed, this would be the first ever democratic change of power in Nigeria and cap a remarkable comeback for the 72-year-old, who headed a short-lived military regime in the 1980s.
With just one state to declare, Buhari is virtually guaranteed victory.
Thousands of jubilant Buhari supporters poured into the streets in celebration, many in northern Nigeria which has borne the brunt of the bloody six-year Boko Haram uprising.
"President Jonathan called at 5:15 (1615 GMT)," his campaign spokesman Shehu Garba said. Asked by AFP if Jonathan conceded, he said: "Yes, and General Buhari has accepted and thanks him for this."
There was no immediate comment from Jonathan, his spokesman or the ruling People's Democratic Party (PDP), but Buhari's wife Aisha took to Twitter to celebrate.
"We see this as a trimphant show of democracy, a change for the better," she wrote.
Bukhari had told AFP he was "very confident" of victory as counting of votes showed him pulling well ahead of Jonathan.
The final result is awaited from Borno state, the heart of the Boko Haram Islamist insurgency.
"Change, change," chanted opposition supporters in the northern city of Kaduna.
In Kano, some young men donned black fedoras -- which his rival Jonathan is rarely seen without -- and put suitcases on their heads to mock the president as people chanted "Out of Aso Rock" (the presidential villa).
Buhari won a landslide victory in Kano, Nigeria's second most-populous state, securing more than 1.9 million votes and 89 percent of the vote.
Buhari was more than 2.75 million votes ahead of 57-year-old Jonathan, after winning in the northern states of Yobe and Adamawa.
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