Dawson, who is out-of-contract at the Swans as a restricted free agent, will require the club to agree to an exchange that will see the 24-year-old join either the Crows or Power.
The Robe product has played 64 games for the Swans since joining the club, making his debut in 2017.
The Swans confirmed the news on Friday afternoon, with club football boss Charlie Gardiner revealing Dawson's desire to head back to SA.
“Jordan has indicated to the club his desire to return to South Australia to be closer to family,” Gardiner said via the club's announcement.
“Jordan has been weighing up this decision for some time and despite there being an attractive long-term deal on offer with the Swans, he has arrived at the decision that he would like to return home.
“The news is obviously disappointing for the club. Jordan is a young, hugely talented player who is coming off a breakout season in 2021, which only adds to the disappointment.
“Given we have only learned the news this afternoon, we will take some time to work through the Club’s options leading into the trade period."
Dawson has emerged as a key asset to the Swans, being able to play across multiple lines and finished third in the club's 2021 Best and Fairest count.
As things stand, Adelaide currently hold picks 4, 23, 37, 59, 63, while the Power hold selections that will land close to 17, 60, 71, 72 depending on this weekend's results.