However, the BBC’s Rob Cameron in Prague says there are indications that the election will go to a second round later this month.
Mr Zeman, 73, is seeking a second five-year term but has stoked controversy with his outspoken views and pro-Russian stance.
Turnout in the first round, which took place over two days, was about 60%, election officials said.
Most of the early results came from rural areas where Mr Zeman’s support is high. To avoid a run-off he must win more than 50% of the vote, but observers say that is unlikely unless there are dramatic changes in urban results.
If a second round is confirmed, it is expected to be held on 26-27 January.
Mr Zeman’s forthright views on immigrants and political correctness have struck a chord in rural communities, our correspondent says.
However, urban voters are more likely to agree with his pro-European challengers, who say he has poisoned the political atmosphere and unsettled allies.
The first round of voting passed uneventfully apart from a semi-naked protester who tried to disrupt Mr Zeman casting his vote in Prague on Friday.
The topless woman from the feminist group Femen accused him of being in the pocket of Russian President Vladimir Putin. She was bundled away and Mr Zeman then cast his ballot, but was visibly shaken by the incident.
President Zeman has become one of the EU’s most outspoken opponents of sanctions against Moscow and has also made improving relations with China a priority.
Mr Drahos, a pro-European academic, has been forthright in his opposition to the president, saying: “We say in Czech that ‘the fish stinks from the head’ and that perfectly sums up Mr Zeman’s term.”