The newly-promoted side have enjoyed a productive transfer window, adding as many as 15 fresh faces to their first-team squad.
However, despite making a promising start to the new Premier League campaign, head coach Steve Cooper is still in the market for several additions.
The recent acquisition of Emmanuel Dennis means that Forest have four recognised forwards in their ranks, even when not considering other players who may depart the City Ground.
Nevertheless, the East Midlands outfit have continued to be linked with a move for Hwang, who has been made available by Bordeaux after their relegation from Ligue 1.
According to transfer expert Fabrizio Romano, Forest are on the brink of announcing the signing of the South Korea international.
While Romano does not mention a fee, he says that the 29-year-old will pen a contract until the end of the 2024-25 campaign, with the first year being spent away from the club.
Olympiacos - who like Forest are owned by Evangelos Marinakis - are in line to sign Hwang on a season-long deal.
The attacker has played just 20 minutes of football in the French second tier this season having scored 23 times during his last two years in the top flight.
Forest winning the race for Hwang would be regarded as somewhat of a coup with other Premier League clubs known to be admirers of his ability.
Wolverhampton Wanderers have been searching for a striker for a number of weeks, while Fulham allegedly viewed Hwang as backup to Aleksandar Mitrovic at one stage.
Other clubs across the continent would have been naturally monitoring the situation with Hwang in the last 12 months of his deal at Bordeaux.
Olympiacos also signed Hwang's compatriot and international teammate Hwang In-beom less than a month ago.