TV star Salvador Nasralla, who claimed victory in the Nov. 26 election after early results put him ahead of Hernandez, has been locked in a bitter row over the vote count since the process broke down and suddenly swung in the president's favor.
The dispute has sparked deadly protests and a night-time curfew in the poor, violent Central American country. On Monday, rebel police refused to crack down on demonstrations, urging the government to resolve the political deadlock.
Salvador Nasralla, presidential candidate of the Opposition Alliance Against the Dictatorship applauds next to Former Honduran President Manuel Zelaya as they attend a news conference in Tegucigalpa, Honduras, Dec. 4, 2017.
On Tuesday, Nasralla said on Twitter the electoral tribunal should review virtually all the voting cards.
'If you don't agree with that, let's go to a run-off between (Hernandez) and Salvador Nasralla,' he said.
Former President Manuel Zelaya, who was ousted in a 2009 coup and now backs Nasralla, said on Twitter that the opposition was seeking a total recount of the vote, or legislation to permit a run-off, which is not used in Honduras.
Authorities took a week to count votes in the nation of 9 million people, but the Organization of American States (OAS) said results were marked by irregularities and errors.
The tribunal has not declared an official winner, but the results gave a 1.60 percentage point advantage to Hernandez over Nasralla, who says tally sheets from ballot boxes were altered and has declared himself the rightful winner.
On Tuesday, the top official at the electoral tribunal, David Matamoros, invited the opposition to compare their copies of voter tally sheets with the official body's versions.
Matamoros also said the tribunal would extend a deadline for legal challenges to Friday from Wednesday.
Street protests in favor of Nasralla that began last week continued on Tuesday afternoon. Dozens of people, including police officers, gathered at the Tegucigalpa headquarters of Honduras' elite police force yelling 'Out, JOH,' referring to Hernandez's initials.
Nasralla's center-left Alliance bloc previously demanded a recount of nearly a third of tally sheets, a request that was backed by the OAS and European Union election observers. The Alliance is also expected to formally contest the results.
Honduras President and National Party candidate Juan Orlando Hernandez gestures as he addresses the media in Tegucigalpa, Honduras, Dec. 4, 2017.
Hernandez, who has been praised by the United States for his crackdown on violent street gangs, claimed victory several times since the election, but did not make that claim in broadcast comments on Monday and Tuesday.
Early last week, Nasralla, a 64-year-old former sportscaster and game show host, appeared set for an upset victory, gaining a five-point lead with more than half of the ballots tallied.
The count halted for more than a day, and began leaning in favor of Hernandez after resuming.