The move comes on the back of a solid start to the campaign for the Bundesliga club, who beat second-tier newcomers Energie Cottbus 2-0 in the first round of the DFB Pokal, with goals from Anton Kade and Kristijan Jakić, ahead of their league opener against Schalke 04 this coming weekend.
According to the Serbian outlet, Augsburg will pay a fixed fee of €4million for the 27-year-old, with a further €1.5million available in bonuses and a 10 per cent sell-on clause included as part of the agreement.
Seol is expected to be deployed primarily on the right side of Augsburg's defence once the move is completed.
The South Korean right-back arrived at Red Star from Ulsan HD and has gone on to make more than 80 competitive appearances for the Serbian giants, contributing several goals and a healthy number of assists from full-back. He has also become a regular fixture in the South Korea national team, earning around 30 caps, and started at this summer's World Cup in North America.
Seol's contract in Belgrade runs out next summer, a factor that has likely accelerated Red Star's willingness to negotiate a fee now rather than risk losing him for nothing in twelve months' time.
Eintracht Frankfurt had also been credited with an interest in the defender earlier this year, but it is Augsburg who appear to have won the race for his signature.
Should the deal go through as reported, Seol would become one of Augsburg's more eye-catching additions of the summer, bringing Bundesliga-ready experience and a proven track record in a competitive European league to Manuel Baum's back line ahead of Sunday's league opener against Schalke 04.
Augsburg finished ninth in the Bundesliga last season and reached the second round of the DFB Pokal, results that underline a club looking to build steadily rather than take major risks in the transfer market.
A defender with Champions League and international tournament experience at a relatively modest fee fits that pattern, particularly given Seol's versatility as an attack-minded full-back capable of contributing at both ends of the pitch.