A former mayor of Tegucigalpa, 67-year-old Asfura had 41 per cent of the votes counted, preliminary results showed on Monday.
The preliminary results showed Liberal Party candidate Salvador Nasralla in second place with about 39 per cent.
Rixi Moncada, of the ruling LIBRE party, had 20 per cent, and was in third place.
Whichever candidate wins a simple majority will govern the country between 2026 and 2030.
In the run-up, Trump weighed in on the tightly contested race to throw his support behind Asfura in a series of social media posts, saying he can work with him to counter drug trafficking and that "if he doesn't win, the United States will not be throwing good money after bad".
On Friday, Trump also said he would grant a pardon to former Honduras President Juan Orlando Hernandez, who is serving a 45-year prison sentence in the US for drug trafficking and firearms charges.
Hernandez, who led Honduras from 2014 to 2022, was also of the National Party.
Sunday's vote, in which the 128 members of Congress, hundreds of mayors, and thousands of other public officials are also being chosen, took place in a highly polarised climate, with the three top candidates hurling accusations of possible fraud.
Moncada has suggested she would not recognise the official results.
Most polls showed a virtual tie between the three candidates heading into election day.
The Organization of American States expressed concerns about the electoral process, and the majority of its members in an extraordinary session last week called for the government of outgoing President Xiomara Castro to conduct elections free of intimidation, fraud and political interference.
US Deputy Secretary of State Christopher Landau also warned the US would respond "swiftly and decisively to anyone who undermines the integrity of the democratic process in Honduras".
Honduras, where six out of every 10 citizens live in poverty, suffered a coup in 2009 when an alliance of right-wing military figures, politicians and businessmen overthrew Manuel Zelaya, the husband of the current president.
In 2021, Hondurans voted massively for Castro, ending more than a century of rule by the National and Liberal parties.
Castro, the first woman to govern Honduras, increased public investment and social spending.
The economy has grown moderately, and poverty and inequality have decreased, although both remain high.
The country's homicide rate has also fallen to its lowest level in recent history, but violence persists.
Human rights groups have criticised Castro for maintaining a prolonged state of emergency in parts of Honduras and for continuing to rely on the military for policing, the approach of her predecessor Hernandez.
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