Prime Minister Robert Golob's liberal Freedom Movement won 29 seats in the 90-member assembly at the weekend while the opposition right-wing Slovenian Democratic Party, or SDS, won 28, according to preliminary results of 99.85 per cent of votes counted.
The outcome means that no party has a clear majority and that a future government will depend on smaller parties that emerged as kingmakers following the vote.
It was not immediately clear what shape potential future alliances might take.
"I urge them to sit down at the negotiating table as soon as possible," President Natasa Pirc Musar said on X on Monday.
She congratulated the "relative winner" of the election, the pro-EU ruling Freedom Movement party, which had a lead of less than one per cent.
Sunday's vote was seen as a key test of whether the EU member nation stays on its liberal course or sways towards the right.
The undecided outcome also reflects deep divisions among Slovenia's 1.7 million eligible voters.
Golob's government has been a strong liberal voice in the 27-nation EU.
SDS leader Janez Jansa is a populist-style politician and a close ally of nationalist Hungarian Prime Minister Viktor Orban.
His return to power would be a boost to Europe's right-wing blocs.
Golob has expressed confidence that his party would form the next government though he acknowledged that "tough negotiations" lay ahead.
Jansa, an admirer of US President Donald Trump, said his party would not want to form a weak coalition government.
He said a "balance of political powers ... based on what we see now, will not provide much stability".
The vote was held after a heated campaign that featured allegations of foreign interference and corruption, further whipping already heightened political tensions between the two opposed blocs.
Slovenia routinely has switched between the right and left-leaning blocks since it broke away from the former Yugoslavia in 1991.
The Alpine nation of two million people became a member of NATO and the EU in 2004.
