Kurz's conservative OeVP party won 31.5 percent of the vote in Sunday's election, near-complete results show.
Kurz's conservative OeVP party won 31.5 percent of the vote in Sunday's election, near-complete results show, and the populist Freedom Party (FPOe) came in third on 26.0 percent.
The Social Democrats (SPOe) of incumbent Chancellor Christian Kern came second with 26.9 percent.
"We have several red lines," FPOe chairman Heinz-Christian Strache said in his first press conference since the vote.
"The interior ministry is a precondition" for government participation, he added.
Strache, 48, said he saw "no reason to rush" the negotiations, which could begin at the end of the week.
"We will not enter into a coalition at any price," the far-right leader said.
The FPOe, whose rise has mirrored that of other populist parties in Europe, is demanding increased border security, Swiss-style direct democracy and economic reforms.
Its leaders have also said Islam "has no place" in Austria.
At 31, Kurz is the world's youngest leader-in-waiting. The conservative is expected to be given a mandate to form a government on Friday.
Kurz on Tuesday had said he expected Austria to play an "active" role in the European Union -- in contrast with the long-held views of the eurosceptic, anti-immigrant FPOe.
Strache did not address Kurz's comments on the EU in his press conference.