The victory would make Mr Kurz Europe's youngest leader.
The People's Party was set to win 30.2%, followed by the far-right Freedom Party (FPÖ) with 26.8% and the Social Democrats with 26.3%.
Short of a majority, Mr Kurz's party could seek an alliance with the FPÖ, which campaigned against immigration.
Immigration has been a dominant issue in the run-up to the vote, and the FPÖ is thought to have its best chance in years of returning to government.
| Update at 12:30 p.m. EDT: The Local - Austria is liveblogging the election here. They note official results will not be announced until October 31st. |