Most recently, Kurz criticized Turkish President Recep Tayyip Erdogan's plan to reintroduce the death penalty for those convicted of plotting to overthrow him, describing it as incompatible with EU accession.
"As EU foreign ministers already made clear following the reprehensible coup attempt in July, the death penalty is incompatible with EU membership. Anyone who introduces the death penalty, slams the door to the EU," Kurz said on Sunday, Austria's Kurir reported.
Since his condemnation of Erdogan's plan to reintroduce capital punishment, Kurz has received death threats from supporters of the Turkish President, Austrian newspaper Kronen Zeitung reported on Monday.
"Now there is a true orgy of hate against the Foreign Minister – the wildest abuse, insults and death threats. On Facebook, one Turk threatened to stick a knife in his neck," Austria's Kronen Zeitung reported.
"President Erdogan has again presented plans to introduce the death penalty in Turkey. The death penalty is a cruel form of punishment, which clearly contradicts European values. Anyone who introduces the death penalty, closes the door to the EU," Kurz wrote on his Facebook page.
"I am no Erdogan follower but the way you drag my country through the mud makes me hateful and aggressive. When I see you, I will stick a knife in your neck so that you can't ever talk like that again," one Facebook user wrote to Kurz, Kronen Zeitung reported.
Austria's Foreign Ministry told the newspaper that intelligence services are conducting an investigation into the threat, while Kurz himself remains calm "because insults and threats have already become routine."
"Following pro-Erdogan demonstrations in Vienna last summer, Kurz suggested to supporters of the Turkish President that they leave Austria," Kronen Zeitung recounted.
Pro-Erdogan protests took place in Turkey and some European cities in the aftermath of the failed military coup in Turkey on July 15. The Turkish government blamed the coup attempt on US-based Muslim cleric Fethullah Gulen, and accused his followers of forming a "parallel state" within Turkey.
On July 25 Erdogan told German broadcaster ARD that the Turkish people want capital punishment reintroduced for the coup plotters.
"What do the [Turkish] people say today? They want the death penalty reintroduced. And we as the government must listen to what the people say. We can't say 'no, that doesn’t interest us,'" Erdogan said.
"Only in Europe is there no death penalty. Otherwise, it is almost everywhere," he added.
On Saturday Erdogan reiterated his commitment to reinstating the death penalty, in spite of opposition from EU politicians.
"Our government will take this (proposal on capital punishment) to parliament," Erdogan said during a speech to inaugurate a high-speed train station in Ankara.
"I am convinced that parliament will approve it, and when it comes back to me, I will ratify it," the President told the crowd, who chanted "We want the death penalty!"