The 19-year-old was in his country's squad for the recent Copa America and the Under 20 World Cup.
The Gunners revealed that the transfer is subject to the completion of 'formal agreements and processes'.
Head boy: Joel Campbell (left) in action at the U20 World Cup
Manager Arsene Wenger told his club's official website: 'Joel Campbell has already shown that he is a player with great ability, and has also performed well on the international stage at a young age.
'We look forward to the formalities of the transfer being completed and working with Joel when he joins us at Arsenal.'
Campbell is predominantly left-footed and has made six full international appearances, scoring twice.
He scored on his debut in a 5-0 win over Cuba at this year's CONCACAF Gold Cup, helping his side to the competition's quarter-finals.
Campbell has had two seasons in his native Primera Division, where he played for Deportivo Saprissa and on loan at Puntarenas, making a total of eight league appearances.
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