He went on to note that he would suggest to Austrian President Alexander Van der Bellen that a snap election take place as soon as possible.
Austrian Vice-Chancellor Heinz-Christian Strache, a key figure of the European far-right, resigned Saturday after explosive revelations from a hidden camera sting, just days before key EU elections.
Strache quit as vice chancellor and party leader earlier on Saturday after the video was released by two German news organisations.
"You can not draw a Russian link to this clearly ugly incident based on the existing recording", Mr Oleg Morozov, a member of the foreign affairs committee of Russia's Upper House and a ruling party lawmaker, told RIA Novosti state news agency.
The Austrian APA agency reports FPOe party sources as saying they want to try to save the coalition with Kurz's centre-right People's Party (OeVP) by replacing Strache as vice-chancellor with Hofer.
After years of efforts to clean up the image of his Freedom Party (FPOe), founded after World War II by former Nazis, Strache has now been disgraced by what he termed a "stupid, irresponsible mistake".
"Even if I didn't express myself publicly at the time, there were many situations that I found hard to swallow", Kurz said of his time in government with the FPOe since December 2017.
This breaking news story is being updated and more details will be published shortly.
Senior figures in the government - a coalition of Kurz's conservatives and the Freedom Party - held a crisis meeting on Friday night.
According to the two newspapers, the video spanned some six hours of drink-fuelled conversation in a villa on the Spanish island of Ibiza between the Austrian politicians and the woman, who claimed to be the niece of a prominent Russian businessman.
Sueddeutsche Zeitung and Der Spiegel said the footage was authenticated by a forensic video expert.
Strache, whose party has a cooperation agreement with Russia's ruling United Russia party, described the sting as a "targeted political assassination" and said it never led to any money changing hands.
They discussed how the woman could support their campaign during the 2017 general election in Austria in exchange for future preferences in getting government contracts in the field of construction. Kurz has so far not spoken publicly.
As well as repeating unsubstantiated rumours about Kurz, Strache also hinted at ways political donations could escape legal scrutiny by going to a foundation linked to the FPOe.