With 88 per cent of ballots tallied, Asfura of the National Party had 40.19 per cent, about 20,000 votes ahead of rival Salvador Nasralla of the Liberal Party, who stood at 39.49 per cent, according to the electoral authority.
Rixi Moncada of the ruling progressive LIBRE Party remained well behind in third place on 19.30 per cent.
About 14 per cent of ballots showed inconsistencies, officials said, and would be reviewed.
The race has received very close attention from the United States, with President Donald Trump backing Asfura.
Deputy Secretary of State Christopher Landau wrote on social media on Thursday night, "The world's eyes, including ours, are on Honduras."
Days before voting began, Trump announced he would pardon a former president of Asfura's National Party: Juan Orlando Hernandez, who had been serving a 45-year sentence in the US after being convicted on drug trafficking and weapons charges.
In his first video message after leaving a US prison on Tuesday, Hernandez thanked Trump for pardoning him and for Trump's "interest in the fate of our nation".
On Saturday, the streets of the capital, Tegucigalpa, remained calm, as election workers across the country continued counting ballots by hand.
Asfura is the 67-year-old former mayor of Tegucigalpa.
His tenure as mayor was plagued by a series of allegations of corruption.
Nasralla is a three-time presidential hopeful and a former popular TV presenter who describes himself as centre-right.
He said in an interview with Reuters that Trump's surprise endorsement of Asfura had flipped the race.
Moncada, trailing behind the two front-runners, is a 60-year-old teacher and lawyer with the LIBRE Party, which has governed Honduras since 2022.
Honduras elects its president in a single round.
The candidate with the most votes wins, even if the margin is narrow or the candidate falls short of a majority.
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