"Ahn has already signed a three-year contract with us, he will stay with us - he wants to continue in Perugia," said Gaucci's son, Alessandro, who is the club's general director.
Nine days ago, Gaucci senior was widely reported as wanting nothing more to do with the Korean after he had knocked out Italy.
For his part, Ahn denied ever saying anything negative about Perugia and clearly he and the Gaucci family have made up over the past few days.
"I have spoken to Ahn on several occasions and he wants to come back, he has a contract with us and we have cleared all the misunderstanding there was," Luciano Gaucci said on Thursday.
During the past week, Ahn has been linked with moves to several clubs, including Rangers, Chelsea and Atletico Madrid.
A former agent of Ahn's is reported this week to have said the Korean did not want to return to Perugia, but Luciano dismissed such talk as "rubbish".
He added: "The player has already signed a contract with us and those people just want to create confusion.
"Most probably because they did not want the player to sign for us as they would have gained more money with another deal.
"In any case we have a deal with the player and if that is not obeyed it would be a breach of contract, thus breaking Fifa regulations."