Mr Simion and his rival – who is currently Bucharest’s mayor – are facing off in the second round of elections.
Exit polls placed Mr Dan comfortably ahead by 54.9% to 45.1%.
However, Trump-supporting Mr Simion has already rejected his apparent loss and said estimates put him 400,000 votes ahead.
Real-time results show Mr Dan with a razor-edge lead with about half the votes counted.
Romania’s last election was annulled after its highest court ruled the leading candidate, nationalist Calin Georgescu, should be disqualified due to claims of electoral interference by Russia.
The result is surprising because in the first round a few weeks ago, 38-year-old Mr Simion, founder of the right-wing Alliance for the Unity of Romanians (AUR), took 40.96% of the vote – almost 20 points ahead.
An opinion poll on Friday had it much closer, but still suggested the two men were virtually tied.
Mr Dan, a 55-year-old mathematician, is running as an independent and has pledged to clamp down on corruption.
He is also staunchly pro-EU and NATO, and has said Romania’s support for Ukraine is vital for its own security.
When voting closed at 9pm local time, 11.6 million people – about 64% of eligible voters – had cast ballots. About 1.64 million Romanians living abroad also took part.
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