The FPOe topped the poll in September's national election for the first time ever, winning almost 29 percent of the vote, but until now had been unable to find partners to form a national government.
After meeting Kickl for about an hour, President Alexander Van der Bellen said he had tasked the far-right leader with trying to form a government with the conservatives given the "new situation".
Talks between the conservatives and Social Democrats and liberals fell apart over the weekend.
"Mr Kickl believes he can find viable solutions... and he wants this responsibility," Van der Bellen said.
"I have therefore tasked him to enter into talks with the (conservative) OeVP with a view to forming a federal government," he said, adding it was "not an easy" decision.
While the far-right party has been part of Austria's government several times since 2000, it is the first time that it will lead coalition talks.
Analysts told AFP that a coalition led by the far right with the conservatives as junior partners is now "highly likely".
'Nazis out'
Hundreds of people protested against the far right outside the presidency at Vienna's Hofburg palace, shouting "Nazis out" and holding up signs reading "We don't want an Austria ruled by right-wing extremists".
Van der Bellen had initially asked the long-ruling conservative People's Party (OeVP), who came second in the September vote, with forming a stable government that respects the "foundations of our liberal democracy".
But on Sunday, Van der Bellen said a "new situation" had emerged, with "voices within the People's Party (OeVP) that rule out working with.... Kickl have become significantly quieter".
In the past, the president has voiced reservations about Kickl.
Kickl, who took over a scandal-tainted FPOe in 2021 and led its recovery, is known for his virulent rhetoric, including slamming Van der Bellen as a "senile mummy".
He has also frequently employed terms reminiscent of the troubled past of the party, founded by former Nazis in the 1950s, including calling himself the future "Volkskanzler" -- the people's chancellor -- as Adolf Hitler was termed.
- 'Wasted months' -
In the wake of the collapse of the coalition talks on Saturday, conservative Karl Nehammer said he would step down as chancellor and party chairman of the OeVP to enable an "orderly transition". He has held both posts since late 2021.
In a U-turn, the conservatives on Sunday under their new interim party leader Christian Stocker said they would enter into coalition talks with the far right if invited to do so.
"This country needs a stable government right now, and we can't keep losing time to election campaigns," Stocker said.
Nehammer said Saturday that he had wanted to be "the force of the political centre in order to build a bulwark against the radicals".
The same day Kickl branded the parties involved in the failed coalition negotiations "losers".
He added that "instead of stability, we have chaos" after three "wasted months".
The conservative People's Party came second with 26 percent of the vote in the September elections, while the centre-left Social Democrats took 21 percent.
The FPOe has never governed the EU country of nine million. It already leads one regional government in Austria and is part of government in four other provinces.
