During their talks they said they looked at what went wrong at the World Cup, admitting they were still hurting from the experience.
“Our discussions began with the question of what needs to change to ensure that we do not repeat the same mistakes,” they said in the statement.
They added: “We now intend to put unnecessary controversy behind us and focus on building a stronger national team together.”
Spanish coach Robert Moreno took over from Hong on an interim basis, with his contract reportedly set to expire this month.
South Korea play Saudi Arabia in Qatar on Friday before another friendly, against Qatar, four days later.
The matches were originally designed as build-up to the 2027 Asian Cup in Saudi Arabia but the tournament was last week postponed to December next year.
