The Taeguk Warriors departed the 2014 World Cup in the first round after earning a solitary point in the group stages.
According to The Guardian, Hong had tendered his resignation after South Korea's 1-0 defeat to Belgium in their final group game, but Korean Football Association president Chung Mong-gyu talked him out of stepping down.
Huh Jung-moo, the KFA's vice president, is reported as saying: "I don't think Hong's resignation would have solved the problem so we decided to continue to trust him.
"Hong said he intended to resign but Chung Mong-Gyu persuaded him to keep his post until the Asian Cup [in 2015].
"Based on the failure at the World Cup, I am sure Hong will lead the team well at the Asian Cup."
South Korea reached the semi-finals of the 2002 World Cup - played on home soil - under the guidance of Guus Hiddink.