The probe could hurt Kurz and his conservative People's Party (OeVP), which is so far tipped to come out strongest again in September elections despite the scandal that has since become known as "Ibiza-gate".
Hidden camera recordings saw Kurz's far-right vice-chancellor, Heinz-Christian Strache, resign, the coalition collapse and a caretaker government appointed.
The recordings filmed on the island of Ibiza in 2017 showed Strache appearing to offer public contracts in exchange for campaign help to a fake Russian backer.
In a new twist, prosecutors are now also investigating a staffer of Kurz's team for shredding a hard disk on May 23, days after the scandal broke and just before a successful no-confidence vote against the chancellor's government.
The staffer, who has not been named by Austrian media, is under suspicion of destroying evidence, according to the Kurier daily on Saturday.
The shredding company informed police after the staffer failed to pay the 76-euro ($85) bill and could not be contacted as he had given a fake name.
Austrian lawmakers, not from Kurz's People's Party (OeVP), have called for a full investigation.
"This whole matter throws up lots of questions which the 'Ibiza' investigators will hopefully clear up soon -- voters have a right to know what scheming OevP chief Sebastion Kurz is involved in," Thomas Drozda of the Social Democrats (SPOe) said in a statement.
The OeVP said it was normal to destroy personal data and non-official documents before a change in government, adding the staffer did not have access to any sensitive information.
"This is a completely normal procedure," an OeVP spokesman told AFP on Sunday.
The spokesman said he did not know why the staffer, who was quizzed by authorities on Thursday and whose home was also searched, had given the shredding company a wrong name.
Kurz's party is the clear favourite to come out strongest in September polls, although it is unlikely to be able to form the next government on its own.
Speculation has been rife that Kurz might be implicated in the Ibiza affair.
Last month, Kurz gave a press conference where he insisted emails connecting him to the scandal were fake. But he did not reveal the content of the emails.