Mr Pahor is a former young communist and one-time male model
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Slovenia's opposition is holding a razor-thin lead over the ruling party of PM Janez Jansa, near-complete results from parliamentary polls show.
With 99% of the votes counted, the Social Democrats had 30.5% of the vote against 29.3% for the centre-right Slovenian Democrats, officials said.
But a strong performance by the Social Democrats' allies gives the centre-left 43 seats in the 90-strong parliament.
Final results are due next week after votes cast abroad are counted.
Slovenia, the richest of the former Yugoslav states, is a member of the European Union and Nato.
It was also the first East European state to adopt euro.
Mr Jansa's party is claiming credit for the country's increased prosperity.
But the centre-right government has also frequently been accused of corruption.
Jansa's 'admission'
The Social Democrats led by Borut Pahor are now expected to get 29 seats in parliament, one more than Mr Jansa's party.
Mr Jansa is hoping to gain a new four-year mandate
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Mr Pahor's main allies - Zares and the Liberal Democrats - received 9.4% and 5.19% of the vote respectively.
The three parties look set to control about 43 seats in parliament - just three short of a majority.
"This is a great achievement for us but we will still have to wait for the official final results by the [electoral] commission," Mr Pahor told reporters on Sunday.
Mr Jansa's allies fared badly in the polls, with one of the centre-right parties failing to reach the 4% threshold.
Mr Jansa admitted that the centre-left coalition had "more chance" of forming the next government.
While the economic policies of the two main parties are similar, a left-leaning government could be expected to focus more on the redistribution of wealth to poorer parts of society, the BBC's Nick Thorpe says.
Some 1.7 million people were eligible to vote.
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