Kickl's party was rocked by the publication of a video last week apparently showing its leader, Heinz-Christian Strache, offering a potential wealthy Russian benefactor government contracts in return for support.
Van der Bellen approved on Saturday the proposal of holding snap elections after Strache resigned over an alleged corruption video and effectively led to the collapse of the coalition government.
Strache, whose party has been one of the most successful nationalist and anti-immigration movements in Europe, securing more than a quarter of the vote and taking the interior, foreign and defense ministries, denies doing anything illegal.
The Freedom Party had said earlier on Monday it would pull its ministers out of the government if Interior Minister Herbert Kickl were forced out in the wake of a video sting that felled its leader at the weekend.
"It is clear Herbert Kickl can not probe himself", Kurz was quoted by the Kurier newspaper on Monday.More news: Maddon, Cubs drop protest over Doolittle's delivery
Since Kickl was General Secretary of the FPO when the video was shot, this could not guarantee a transparent investigation, said Kurz. Strache said that a number of staff journalists critical to the FPO at Krone Zeitung would need to be fired and others, more sympathetic to FPO (evidently) brought in.
The FPO had announced that it would vacate all its ministerial posts if Kickl, a mastermind of the FPO's ascent to power, were forced out.
He said the recordings were the final straw in a string of FPOe-related scandals.
Austria is preparing for an election in September after, what is being labelled, the "Ibiza scandal".
Kickl, a longtime campaign mastermind of the Freedom Party, has drawn criticism over matters including a raid past year on Austria's BVT spy agency, which opposition parties claimed was an attempt by the new government to purge domestic political enemies. "We do not know who this woman is, whether she is Russian or not".More news: Man charged with assaulting Brexit Party leader Nigel Farage after milkshake incident
Kurz has endorsed a hard line on migration and public finances, and chose to ally with the Freedom Party after winning the 2017 election.
In 2016, the FPO signed a cooperation agreement with Putin's United Russia party.
The Kremlin shrugged off suspicions of involvement in the scandal.
The scandal may also dent the prospects of the far-right populist alliance marshalled by Italy's Interior Minister Matteo Salvini, in which the FPOe plays a key part.More news: Maisie Williams Admits Her One 'Game Of Thrones' Final Season Regret