Kurz's conservative ÖVP party won 31.5 percent of the vote in Sunday's election, near-complete results show, and the populist Freedom Party (FPÖ), came in third on 26.0 percent.
The Social Democrats (SPÖ) of incumbent Chancellor Christian Kern came second with 26.9 percent.
“We have several red lines,” FPÖ 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 are to due to begin by the end of the week.
“We will not enter into a coalition at any price,” the far-right leader said.
The FPÖ, 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 due to be sworn in on Friday and could then be given the go-ahead to begin holding coalition talks with all parties.
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 FPÖ.
Strache did not address Kurz's comments on the EU in his press conference.