The 19-year-old was in his country's squad for the recent Copa America and the Under-20 World Cup.
The Gunners revealed on Friday night that the transfer is subject to the completion of `formal agreements and processes'.
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.