Italian Prime Minister Matteo Renzi last week insisted Italy was not facing an "invasion" after a spike in migrant boat crossings from Libya exacerbated fears it could become the main entry point into Europe.
But Kurz highlighted Austria's deep concerns as Europe struggles to contain the worst migrant crisis the continent has known since World War II.
"Look at the figures. Austria received 90,000 migrants last year, close on one percent of the population - in terms of Italy's population that would amount to more than 600,000 people," Kurz told Italian daily Il Messaggero.
He noted that 154,000 migrants had arrived in Italy across 2015 but had "only" registered 83,000 asylum claims - a discrepancy of 71,000.
Austria, seeking to cut this year's arrivals to 37,500, sparked consternation in Rome last week over preliminary construction work at Brenner Pass in the Alps to prepare for a possible new influx of migrants coming north from Italy.
The Brenner Pass is one of the main transport corridors between northern and southern Europe.
The European Union also last week echoed Italian concern, with European Commission spokeswoman Natasha Bertaud saying the issue would need to be discussed with Austrian Interior Minister Johanna Mikl-Leitner given the Pass is "essential for freedom of movement within the European Union."
Vienna worries that the closure earlier this year of the Balkan trail from Greece towards Austria means a new route across the sea from Libya to Italy and then northwards will open up.
Switzerland revealed its own concern at the crisis, saying it was ready to mobilise up to 2,000 soldiers should migrants continue to arrive.
Swiss federal authorities said they have drawn up separate scenarios ranging from 10,000 arrivals inside a month to a worst case of 30,000 inside a few days.
The latter "would necessitate army intervention," the government said.
In a statement the government said it had tasked the defence ministry with putting 2,000 troops on standby for such an eventuality.
The ministry last week said that given the current shifting state of migratory routes "Switzerland could be faced in the coming weeks or months by an influx of people seeking protection."
Earlier this week, Italy's Coldiretti agricultural union said the reintroduction of border controls would threaten the movement €10 billion ($11.3 billion) worth of Italian foodstuffs which transit the tunnel annually.
"We are aware of the historic and logistical importance of this border and we ourselves do not wish to see controls," Kurz said.
"But if we find the number of clandestine migrants arriving via the Mediterranean does not drop considerably then we would be forced to introduce controls at Brenner," he warned.