The Costa Rica international was expected to complete a move to the Emirates Stadium from Deportivo Saprissa within the next few days.
The 19-year-old has, however, opted to postpone a move to England, choosing instead to concentrate on the Under-20 World Cup which ends in August.
A number of European clubs are thought to be interested in the player, with Fiorentina the latest to make a firm offer.
"He has lost a great opportunity [turning down Arsenal]," FIFA agent Joachim Batica, who is currently working for Deportivo, told La Nacion.
"From the outside he seems to be passing up the best opportunity of his life, but I still don't know how the matter will end.
"I will do my best to give Joel other opportunities. I will continue working to see if there is any chance of re-opening negotiations.
"There is also an offer from Fiorentina for Campbell. Joel is aware of the offer and is discussing it with his family.
"It is a good deal, both from a sporting and economical perspective."
Campbell has made six appearances for the Costa Rica national side.