"We must resolve the migrant question and if there is no solution at the (EU leaders') June summit, then we will have to continue to tackle the issue," Kurz said after talks in Brussels with European Commission head Jean-Claude Juncker.
"We will concentrate on this issue at the leaders summit September 20th in Salzburg with the aim of taking another step towards protecting the EU's external frontiers," he said.
Austria takes up the European Union rotating presidency on July 1 and its focus on controlling immigration will go down well with like-minded nationalist governments coming to power in the 28-nation EU, the latest in Italy.
The new Italian government said earlier this week it wanted to stop having to take in thousands of refugees crossing the Mediterranean, insisting the EU should do more to help and that economic migrants should be sent back to North Africa.
Under the EU's current Dublin rules, asylum seekers are supposed to be processed in the first country they land in but this has left Italy and Greece bearing the brunt of the problem.
EU leaders meet June 28-29 in Brussels supposedly to agree changes to the Dublin rules but talks this week failed to make much progress.
The principal sticking point is that many countries, especially in eastern European states such as Hungary and Poland, refuse to take in EU-set quotas of refugees for processing to ease the pressure on Italy and Greece.
"There will certainly be lively discussions on the quotas," Kurz said, adding there was general agreement on tightening control over the EU's external frontier so as to reduce the numbers of migrants coming to Europe.
Juncker said "we must find a solution ... when that is possible."
"If we can make progress on the frontier issue, the other questions should become secondary," said Juncker, who backed the quota system which was introduced at the height of the migrant crisis in 2015.