Austria was in the minority of EU countries that did not expel any Russian diplomats over the poisoning of former spy Sergei Skripal and his daughter in Britain, which London has blamed on Moscow.
But Kurz hardened his tone on Friday, announcing that a recently retired colonel was believed to have spied for Moscow from the 1990s until this year.
In Moscow, Russian Foreign Minister Sergei Lavrov said he was "unpleasantly surprised" by Vienna's allegations and Moscow knew nothing about the retired officer, news agencies RIA and Interfax reported.
Kurz repeatedly referred to a recent case in the Netherlands in which the government said Russian agents tried to hack into the Organisation for the Prohibition of Chemical Weapons while it was investigating the Skripal case.
"Based on the information we have but also because of recent events for example in the Netherlands we can very much assume at the moment that our suspicion will be confirmed," said Kurz, who has met Putin in Austria twice since the Skripal poisoning.