Stiles was fired last week after coaching the Reds to just four wins in his first season solely in charge of the Brisbane-based Super Rugby team.
Ledesma, who was credited for fixing the Wallabies' scrum before their run to the 2015 World Cup final, is returning to his native Argentina to coach the Jaguares.
Stiles said he felt he still had a lot to offer Australian rugby.
"I think what I was able to achieve as a forwards coach at both the (Western) Force and the Reds is proof of that," the 43-year-old former Wallabies prop told the Australian Rugby Union website (rugby.com.au).
"If the opportunity was to come up, I'd love to be part of what the Wallabies are building.
"They're clearly on their way to bigger and better things and I feel like I could contribute to that moving forward."
The Wallabies defeated Argentina away on Saturday to finish second behind the All Blacks in the Rugby Championship for a second successive year.
They play the world champion All Blacks in the third and final Bledisloe Cup test in Brisbane on Oct. 21.
The All Blacks took an unassailable 2-0 lead in the trans-Tasman series with victory over the Wallabies in Dunedin, retaining the trophy they have held since 2003.
(Reporting by Ian Ransom; Editing by Greg Stutchbury)
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