"All members ... agreed to name a Korean coach," Lee Hoi-taek, chief of the Korea Football Association (KFA) technical panel, told reporters on Wednesday.
"We are looking into some 12 or 13 football leaders on the list."
Chung Hae-soung, Huh's chief assistant, and Olympic coach Hong Myung-bo have both ruled themselves out of the running.
Kim Hak-beom, former coach of seven-times K-League champions Seongnam Ilhwa, and Choi Kang-hee, manager of 2009 K-League champs Jeonbuk Motors, now head the shortlist, according to South Korea's Yonhap news.
"Considering Huh's accomplishment (in South Africa), the next coach must be as competitive and experienced as Huh," Lee said.
"The next leader will be in charge of the national team until the 2014 Brazil World Cup."
South Korea reached the semi-finals of the 2002 World Cup as co-hosts under Dutchman Guus Hiddink but failed to capitalise on his success under four subsequent foreign coaches.
The 55-year-old Huh steadily won over his critics after initially being accused of playing "nihilistic football" in South Korea's hypercritical media.
After sealing South Korea's seventh successive World Cup appearance in an unbeaten qualifying campaign, Huh admitted his side were some way off reproducing their astonishing 2002 run.
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