"I have started with the expectation that I will win this election," Turk said on Sunday.
Lojze Peterle, a former prime minister and the ruling centre-right coalition's candidate, is tipped to secure between 32 per cent and 31.1 per cent of the vote, according to opinion polls.
"I expect tight but positive results," Peterle said after voting in Ljubljana's residential district of Brezice.
Peterle won 28.73 per cent support in the first round of voting, ahead of Turk’s 24.47 per cent.
Opinion polls predicted that Turk, backed by the main opposition centre-left Social Democrats party, would gain votes that went in the first round to Mitja Gaspari, a former central bank governor.
A victory for Turk would be a setback for the centre-right coalition of Janez Jansa, prime minister, before 2008 general elections.
The coalition was defeated in Slovenia's two largest cities, Ljubljana and Maribor, in municipal elections last year.
Charges and denials
Jansa has accused Turk of representing Yugoslav interests during the country's attempt to secure independence in 1991.
Turk has denied the accusations and said he would work with the government if he won the presidential vote.
Sunday's winner will replace Janez Drnovsek, who decided not to run for re-election after being diagnosed with cancer.
Turnout in the first round was 57.7 per cent, down from 72 per cent during the last presidential elections in 2002.
Slovenia, which has a population of two million people, joined the European Union (EU) in 2004 and adopted the euro currency in January.
The new president will head the former communist state when it takes over the revolving six-month presidency of the EU in January.