On Friday Germany's Der Spiegel and Sueddeutsche Zeitung published hidden-camera recordings of a sting operation they say took place a few months before 2017's parliamentary elections in Austria.
In them Strache and his party colleague, MP Johann Gudenus, are seen discussing with a woman purporting to be the niece of a Russian oligarch how she can invest money in Austria, specifically in order to gain control of the country's largest-circulation tabloid, the Krone Zeitung.
In the recordings, Strache suggests that under new leadership the Krone could help his Freedom Party (FPOe) in its election campaign.
He goes on to suggest the woman would then be able to gain access to public contracts.
None of the publications have disclosed who filmed the meeting or who set up the elaborate sting, filmed in a villa on the island of Ibiza.
Strache admitted the meeting took place but denies any wrongdoing.
He told the Sueddeutsche Zeitung that "a lot of alcohol was consumed as the evening progressed" and that there was a "high language barrier" during the conversation.
The main opposition Socialist Party (SPOe) demanded Strache's resignation, calling the allegations "the biggest scandal" of Austria's post-war republic.
The liberal NEOS party echoed the call and also said that fresh parliamentary elections were now "unavoidable".
The latest allegations, just days before key EU elections, follow a tense period for the Freedom Party's coalition with the centre-right People's Party of Chancellor Sebastian Kurz.
Kurz has had to defend his decision to enter government with the far-right after a string of revelations about extremist sympathies in the FPOe's ranks.