"We will give up our government offices if Interior Minister Herbert Kickl is forced out," said Norbert Hofer, who is infrastructure minister and head of the far-right Freedom Party (FPOe).
On Sunday Austrian President Alexander Van der Bellen said Sunday he wanted snap elections to take place in September after the government collapsed over a corruption scandal.
The Kremlin on Monday said the corruption scandal involving Austria's far-right party that caused the government to collapse has nothing to do with Russia.
"This is an incident that has not and could not have anything to do with us," Kremlin spokesman Dmitry Peskov told journalists in Russia's first official comment on the scandal.
On Friday far-right Vice-Chancellor Heinz-Christian Strache resigned in disgrace following explosive revelations in a hidden camera sting.
On Friday, two German newspapers published footage showing Strache openly discussing the possibility of awarding public contracts in return for campaign help for his far-right Freedom Party (FPOe).
Strache stepped down as vice-chancellor and FPOe leader after recordings published by German media Friday showed him offering government contracts in return for campaign help to a fake Russian backer in a villa on the resort island of Ibiza.
Elsewhere in the footage, Strache appears to hint at ways political donations could escape legal scrutiny.
Kickl was FPOe secretary general at the time when any political donations would have been made. Strache on Saturday denied the party had received illegal funds.
"It is clear Herbert Kickl cannot probe himself," Kurz was quoted by the Kurier newspaper on Monday.
He said the recordings were the final straw in a string of FPOe-related scandals.
Perhaps the most damaging recent controversy linked to interior minister Kickl was last year's raids on the domestic intelligence agency BVT.
Numerous documents were seized, raising fears among Austria's Western partners about the possibility of leaks to Moscow.
The FPOe has a cooperation agreement with President Vladimir Putin's United Russia party.