Hong's position came under scrutiny after South Korea, who reached the semi-finals in 2002 on home soil, finished bottom of their group and earned just one point. It was the first time they had failed to reach the knockout stage since 1998.
Huh Jung-moo, the Korean Football Association vice president, said Hong had tendered his resignation after Korea's 1-0 loss to Belgium in their final group game but KFA president Chung Mong-gyu had talked him out of stepping down.
"I don't think Hong's resignation would have solved the problem so we decided to continue to trust him," Huh said.
"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 have been drawn in the same group as hosts Australia for the January 9-31 Asian Cup next year. Hong's contract expires at the end of the tournament.
Hong, who led the South Korean Olympic team to a bronze medal in London, has only been in the job a year after taking over from Choi Kang-hee at the end of a woeful World Cup qualifying campaign.
KFA Vice President Huh said the association had to shoulder much of the blame for the team's poor showing in Brazil after giving Hong just 12 months to prepare.
"The preparation period was short and insufficient," continued Huh. "Just because he is the head of the national squad we don't think it's right for him to take all the responsibility. We will continue to support and trust him."
South Korea's most capped player with 136 appearances, Hong skippered the Guus Hiddink-coached team on their run to fourth place in 2002, the best finish by any Asian team at a World Cup.
After taking the helm of the national team, Hong tinkered with formations and personnel but could not get his team firing on all cylinders in the build-up to Brazil.
His decision to stick with misfiring striker Park Chu-young in Brazil drew criticism from fans, while his failure to sort out Korea's defensive woes led to an embarrassing 4-2 loss to Algeria that virtually eliminated them from the tournament.
Korea's only point came from a 1-1 draw in their opening Group H game against Russia.
When the squad arrived back home on Monday they were pelted with toffees, an insult in Korea, and met with fans holding a placard declaring "South Korean football is dead".
Much of the public anger over Korea's failings in Brazil, however, has been channelled towards the KFA and a poll released on Thursday showed 52 per cent of respondents said Hong should remain in the job, with 31 per cent saying he should resign.
South Korea was not the only Asia region team to struggle in Brazil. The other three AFC sides - Japan, Australia and Iran - also finished bottom of their groups and exited the tournament in the first round without a victory among them.
Alberto Zaccheroni has since stepped down as coach of Japan and Iran coach Carlos Queiroz also quit.
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