Jones, a proud Aussie, is seen as an attractive prospect by RA senior figures, who reportedly believe there is too much Kiwi influence in the Australian game, as a potential replacement for Rennie.
RA sees Jones as an improved coach since he was sacked by the Wallabies in 2005, two years after leading the team to a World Cup final.
Jones was briefly linked to a return to the Wallabies following Michael Cheika's resignation after the team's 2019 World Cup quarter-final defeat to England, but RA had already approached Rennie for the job.
"I had some informal contact but nothing too formal," Jones told the ABC at the time. "I'm committed to a two-year contract to England. It would have been difficult to get out of that contract. I think Australia have already moved in other directions."
Asked at the time if there was any genuine interest in the role, Jones said: "Not really. I think it was a convenient conversation for the ARU to have to be honest to show they did show some interest but I don't think they were really interested in bringing me back as Australian coach."
According to the report, RA will continue to honour Rennie's contract regardless of how the team performs ahead of next year's World Cup.
Jones has recently come under fire at his job in England, especially after the first test defeat to the Wallabies in Perth before the visitors turned it around in the next two tests to claim the Ella-Mobbs trophy.
"I love my mother ringing me up in the morning saying 'are you going to get sacked?'," Jones said earlier this month. "My poor mother."
England's tour of Australia ended on a sour note after Jones was called a "traitor" by multiple Wallabies fans at the SCG over the weekend, including an incident where Jones confronted a fan at the stadium.
RA chairman Hamish McLennan said the coach deserved more respect from fans.
"Eddie Jones is a great Australian coach and has made a massive contribution to the game," McLennan told 2GB on Monday. "At the time the [Australian Rugby Union] decided not to give him a job, he's plying his trade overseas, good on him. We should be proud of what he's done."