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The 64-year-old senator has so far won 56% of the vote, compared to 28% for rival Leni Robredo.
Mr Marcos' victory would return his family to power 36 years after they were ousted by a popular revolution.
Critics allege his campaign was fuelled by misinformation, which he denies.
Turnout on voting day was high and previous elections in the country have largely been ruled to be fair. Isolated incidents of violence - including the shooting of three people near a polling station - were not reflected widely across the country, officials said.
But the BBC's Howard Johnson in Manila says there are lingering questions about broken machines at polling stations and videos allegedly showing vote buying.
Opinion polls in the run-up to the election put Mr Marcos Jr ahead of his nearest rival, Ms Robredo, by dozens of percentage points.
Critics say this was the result of Mr Marcos Jr consistently painting his father's rule as a "golden age" for the Philippines, whitewashing a period of rampant corruption and widespread poverty.
His father, Ferdinand Marcos, who became president in 1965, imposed martial law in 1972 and presided over a brutal regime which saw thousands of dissenters and critics jailed and killed.
Mr Marcos Sr, who died in 1989, and his wife Imelda stole an estimated $10bn (£8.1bn) from the Philippines' coffers, becoming infamous examples of public graft.
There was a social media campaign to rebrand the old Marcos era, not as the period of martial law with its terrible human rights abuses, corruption and near-economic collapse, but as a golden age of crime-free prosperity.
This began at least a decade ago, with hundreds of deceptively-edited videos being uploaded to YouTube, which were then reposted on sympathetic Facebook pages.
These have persuaded millions of Filipinos that the vilification of the Marcoses after their downfall was unfair, that the stories of unrivalled greed were untrue.
And then there are the myths, widely believed in poorer parts of the Philippines, that the Marcoses do indeed hold vast wealth in offshore accounts or hidden stashes of gold bullion, but that these are being kept to benefit the Filipino people once they are restored to power.
But the pro-Marcos disinformation campaign also benefited from widespread public disappointment with the failure of the post-1986 administrations to bring significant improvements to the lives of poorer Filipinos.
Bongbong Marcos successfully portrayed himself as the candidate for change, like his predecessor Rodrigo Duterte, promising happiness and unity to a country weary of the years of political polarisation and pandemic hardship and hungry for a better story.
https://www.bbc.com/news/world-asia-61381594
