With votes from just more than 16 percent of ballot boxes counted, Castro had 53.4 percent support, while Asfura had 34 percent, the national electoral council said.
If the opposition standard-bearer wins, she would become the first female president in Honduras and return the left to power for the first time since her husband, former President Manuel Zelaya, was overthrown in a 2009 coup.
The electoral council earlier said more than 2.7 million voters had already cast ballots, a figure described in a statement as a “massive turnout” with more votes yet to be counted.
The initial turnout is already higher than the 2017 total, said council president Kelvin Aguirre. But nearly 8 percent of 5,755 polling places were having transmission problems filing vote tallies with electoral authorities, which was expected to delay results.
A strong turnout has raised expectations of change after a dozen years of National Party rule.
Leftwing Castro has sought to unify opposition to outgoing President Juan Orlando Hernandez, who has denied accusations of having ties to powerful gangs, despite an open investigation in the United States linking him to alleged drug trafficking.
After allying with the 2017 runner-up, a popular TV host, most polls have reinforced her frontrunner status.
“We can’t stay home. This is our moment. This is the moment to kick out the dictatorship,” said Castro, mobbed by reporters just after voting in the town of Catacamas.
Long queues could be seen at many polling places across the country, where some 5.2 million Hondurans are eligible to vote.
The election is the latest political flashpoint in Central America, a leading source of US-bound migrants and refugees fleeing chronic unemployment and gang violence. Honduras is among the world’s most violent countries, although homicide rates recently have dipped.
Central America is also a key transit point for drug trafficking, and where concerns have grown over increasingly authoritarian governments.
The vote also has prompted diplomatic jostling between Beijing and Washington after Castro said she would open diplomatic relations with China, de-emphasising ties with Taiwan, the self-ruled island that China claims as its own.
‘This is Honduras’
Castro’s main rival among 13 presidential hopefuls on the ballot is the National Party’s Asfura, a wealthy businessman and two-term mayor of the capital, Tegucigalpa, who has tried to distance himself from the unpopular incumbent. He has sought to portray his rival as a radical.
After casting his ballot, a measured Asfura said he would respect voters’ verdict.
“Whatever the Honduran people want in the end, I will respect that,” he said.
Some voters consulted by the Reuters news agency expressed dissatisfaction with their choices, but many others had clear favourites.
“I’m against all the corruption, poverty and drug trafficking,” said Jose Gonzalez, 27, a mechanic who said he would vote for Castro.
Hernandez’s disputed 2017 re-election, and its ugly aftermath, looms large. Widespread reports of irregularities provoked protests claiming the lives of more than two dozen people, but he rode out the fraud claims and calls for a re-vote.
Alexa Sanchez, a 22-year-old medical student, lounged on a bench just after voting while listening to music on her headphones and said she reluctantly voted for Castro.
“Honestly, it’s not like there were such good options,” she said, adding she was highly sceptical that the vote would be clean.
“I don’t think so,” she said. “This is Honduras.”
Numerous national and international election observers monitored Sunday’s voting, including the European Union’s 68-member mission.
Zeljana Zovko, the chief EU observer, told reporters around midday that her team mostly saw calm voting with high turnout, although most polling stations they visited opened late.
“The campaign has been very hard,” said Julieta Castellanos, a sociologist and former dean of Honduras’s National Autonomous University, noting that Castro has “generated big expectations”.
Castellanos said post-election violence was possible if the race proved especially close, if a large number of complaints were lodged, or if candidates declared themselves victorious prematurely.
Alongside the presidency, voters are also deciding the composition of the country’s 128-member Congress, plus officials for some 300 local governments.
In Tegucigalpa’s working-class Kennedy neighbourhood, 56-year-old accountant Jose, who declined to give his surname, said he would stick with the ruling party.
“I have hope Tito Asfura can change everything,” he said, using the mayor’s nickname.
“Look, here the corruption is in all the governments.”