Josipovic, a law expert and music composer, won the first round of voting on December 27 with 32 per cent.
Bandic, who is the mayor of Croatia's capital, Zagreb, received about half that number.
Earlier, opinion polls had predicted Josipovic to win the second round with 55 per cent of the votes against Bandic's 38 per cent. But there remain many undecided votes.
Both candidates have pledged to back Croatia's bid for EU membership, which it hopes to achieve in 2012.
No candidate won the more than 50 per cent of first round votes needed to prevent a run-off.
The poll took place amid a deepening economic crisis in the country and concerns among Croats over high-level corruption.
Twelve candidates were in the running in the first round to replace Stipe Mesic, a popular reformist who started the process of getting Croatia into the EU.
Mesic, who led his country into the Nato military alliance, is ineligible to stand for re-election, having already served two terms of five years.
The domestic powers of the Croatian presidency are limited and largely ceremonial, with the main decisions being made by the cabinet and parliament.
But the president leads the army and wields strong influence over foreign policy.
The government has recently begun cracking down corruption, a key condition for joining the EU.
Several of the candidates running in the first round stand accused of questionable past dealings, including Bandic.