Mr Speirs, who quit the leadership a month ago, denies he's the person in a video and a picture published by News Corp.
He said the person in the video was "certainly not me" and called the footage "very troubling".
He released a statement saying: "I believe this is a deep fake or an elaborate hoax".
Mr Tarzia said the video was "a matter for Mr Speirs, and if it is fake, Mr Speirs should refer it to the appropriate authorities".
Vincent Tarzia says it would be inappropriate to comment further until all facts are known. (Sam Wundke/AAP PHOTOS)
"Any suggestion of this kind of activity is of concern to me, and my party takes these matters very seriously," he said.
"It would be inappropriate for further comment until all facts are known."
Mr Tarzia won a partyroom ballot in August after Mr Speirs stepped down, saying he'd "had a gutful" and didn't have the energy to continue as leader.
The Advertiser reported the video was recorded at 4.12am on June 30, while Mr Speirs was still party leader, in the kitchen of his Kingston Park home in Adelaide's southern suburbs.
In a still image, Mr Speirs is captured at the same bench next to a plate with seven lines of white powder. The Advertiser said metadata showed that the image was taken at 6.32am.
The Liberal MP said he "would never have done that" and was "quite horrified".
The newspaper did not cite a source for the video and image but said it had been obtained after it was passed to a third party.
Premier Peter Malinauskas said deepfakes were "a threat to democracy".
"The issue of AI-generated images and deep fakes is a challenge that we've got to confront, and it's not unrelated to the social media challenge," he told Sky News.
"It does represent a threat to democracy. We're going to keep our eye on it and here in the state parliament we're actually bringing in legislation to address it and make sure it's a criminal offence."
Mr Speirs returned last week from Scotland, where he attended a family wedding.
He said he was surprised he was not named in Mr Tarzia's shadow cabinet and has not yet decided if he will recontest his seat of Black at next year's state election.
