Strache said in a statement posted on his Facebook page that the three were "possible accomplices" in the making of the video, in which he appeared to offer public contracts to a fake Russian investor in exchange for campaign help for his far-right Freedom Party (FPOe).
The FPOe has complained that it was illegal for Strache and his party colleague Johann Gudenus to be filmed without their knowledge.
Strache said he would be seeking "the most extensive damages possible" for the "publication of fragments from this illegal secret recording".
Hundreds of FPOe supporters meanwhile rallied in Vienna waving party flags ahead of EU elections on Sunday.
Rosi Imre, 59, a retired cleaner, said she hoped voters wouldn't abandon the FPOe which has won support on an anti-immigration platform.
"FPOe voters should continue to vote. The video is a fake," she claimed.
Another FPOe voter, Walter Kucera, 62, also a retiree, said the video, whose authenticity Strache has confirmed, was a "revenge act" by FPOe's political opponents.
Strache did not give the names of those targeted in his complaints, nor the exact nature of the crimes he alleges were committed.
The explosive video was filmed as part of an elaborate set-up in a luxury Ibizan villa a few months before Austria's last parliamentary elections in 2017. Two German media outlets published extracts from the video last Friday, touching off a crisis which forced Strache from office and prompted Chancellor Sebastian Kurz to end his coalition government. The so-called "Ibiza-gate" scandal also led Kurz to call early parliamentary elections.
But on Monday Kurz himself faces a no-confidence vote, which could see him removed from office in the run-up to the autumn poll.
However, it is unclear if the motion will attract the support from opposition parties necessary to succeed.
Austrian authorities are already investigating possible wrongdoing in the way the video was recorded.
Strache and other FPOe figures may also themselves be targeted by investigators over the deal he allegedly offered to the fake Russian investor and over his comments in the video on party financing.
In the video Strache appears to suggest that donations to the FPOe could be channelled via a foundation linked to the party in order to avoid legal scrutiny.