The Sydney Morning Herald claims senior Rugby Australia officials are telling prospective staff members that New Zealand-born Rennie is set to succeed Michael Cheika following the Wallabies' World Cup quarter-final exit at the hands of England.
Scott Johnson, formerly the SRU's Director of Rugby, is understood to have told Wallabies staff in the aftermath of the 40-16 loss to Eddie Jones' team that Rennie would be taking the job.
Rugby Australia chief executive Raelene Castle was rumoured to be meeting with Rennie's agent in Tokyo this week but insisted on Monday that no appointment had been made.
Speaking earlier this week Ms Castle - who became embroiled in a heated spat with Cheika at a function last month - insisted that Rugby Australia had a "list of people" they would work through in a bid to identify a new head coach.
"We have been doing a lot of work to make sure we understood, from a market perspective, who we think the right coaches could be for Rugby Australia and who might be available.
"That work has been done and continues to be done and so that will be part of the review process."
Despite Ms Castle's comments, Rennie has been linked with the post for some months, and Cheika's resignation in the wake of the Wallabies' World Cup exit added fuel to the fire.
The former Leinster and Stade Francais head coach will leave his post at the end of the calendar year. The highly-regarded Rennie, who led the Chiefs to two Super Rugby titles, is well thought-of in international circles and met with Ms Castle and RA high performance chief Ben Whitaker in November last year.
Glasgow assistant coach John Dalziel told the BBC this week that he could understand why there was interest in the Kiwi.
He added. "There are a lot of articles, a lot of speculation and a lot of theories, but Dave is committed to Glasgow until at least the end of the season.
"There is never a thought of that changing. He is a man of real good stature and he’s working harder than ever to make sure Glasgow have a fantastic season this year and beyond."
Dalziel continued: "He is a very, very highly respected coach all around the world and you can see the effect he has had on Glasgow and we all hope that will continue.
“There is a lot of speculation and that’s all it is at the moment. I know from working with Dave every day that he’s fully focused on the job in hand."