It would be an "enormously challenging task", Schallenberg said, "but I think we are showing an incredible degree of responsibility for this country".
Schallenberg has been Austria's foreign minister since 2019, but he is a relative political novice, Euro News reported.
He has already met with President Alexander Van der Bellen and is set to be sworn in on Monday.
Speaking on Sunday evening, Van der Bellen said the work of the government would continue.
Kurz stepped down on Saturday after prosecutors began investigating him and close team members over claims that government money was used between 2016 and 2018 to ensure positive media coverage.
ZZ/PR