The daily Al Dia said that the London club's scout for the Concacaf region, Richard Law, had arrived in the Central American country accompanied by the agent Joaquim Batica, supposedly with the aim of sealing a deal for Campbell, who plays for Deportivo Saprissa in San Jose.
"Once we have spoken with Saprissa, we will be able to give more information," Batica was quoted as saying.
Campbell started all three of a young Costa Rican side's games at the Copa America, and scored in their 2-0 win over Bolivia.
The player, who will play for his country in the World under-20 Championship in Colombia later this month, has not played in 75 percent of Costa Rica's matches in the last two years, and would therefore not qualify for a work permit to play in England.
However, Batica said that Arsenal may apply to the FA for special permission in order allow a deal to go through.
What the stars mean:
★ Poor ★ ★ Promising ★★★ Good ★★★★ Very good ★★★★★ Exceptional