Cricket Australia CEO Todd Greenberg is in talks with the spinner, who ended his Indian Premier League (IPL) career last month, over a possible stint Down Under this season, according to Cricbuzz.
"To get someone with Ashwin's credentials to come over here for the BBL will be great at so many levels," Greenberg told the website.
"He's a champion cricketer, who will bring a lot to the Big Bash and to our cricket summer."
Ashwin won two IPL titles with Chennai Super Kings before quitting the world's richest T20 tournament last month, saying he would become an "explorer of the game around various leagues".
While several members of the Indian women's team play in the BBL, the country's board does not allow its contracted male players to play in overseas leagues.
Ashwin, who turns 39 this month, was India's frontline test spinner before he quit international cricket last year.