State officials have imposed a lockdown in the district of Guetersloh, after 1,553 workers at Toennies meat processing plant tested positive for Covid-19.
Kurz said a travel warning level 5 has been put in place for the German state.
This is a “partial travel warning to a region," according to the Austrian Interior Ministry travel advisory page.
It advises Austrian citizens against traveling to the region and for any citizens who live there to urgently leave.
The Italian region of Lombardy falls into the same category, according to the ministry.
Kurz was speaking at a scheduled government press conference Wednesday in Austria's capital Vienna.