Bruce wants to take a break from club management following the end of his 2½-year reign at Sunderland.
Bruce was sacked by the Mackems this month, despite guiding them to only their third top-10 finish in the top flight last season in more than 50 years.
Although Bruce retained the support of influential director Niall Quinn, the club’s American owner, Ellis Short, decided a change was needed to stop the team sliding into an unexpected relegation battle.
Short’s close business links with Korea have opened the door for Bruce to possibly become national coach.
The billionaire has spoken highly of Bruce and introduced him to key members of the country’s Football Association during a trip to the Far East in October. Bruce is expected to travel to Seoul to discuss the position early in the new year.
Korea parted company with head coach Cho Kwang-Rae earlier this month following a shock 2-1 World Cup qualifying defeat by Lebanon and are now keen to appoint a foreign coach.
South Korea have enjoyed success in the past with foreign help, most notably the Dutchman Guus Hiddink, who guided them to the semi-finals of the 2002 World Cup, but also his countrymen Dick Advocaat and Pim Verbeek.