Martino is reported to have informed close colleagues of his desire to leave the Catalan club, according to Spanish television show EsportClub.
However, Arroyo hit out at the rumours and scotched any talk of an exit for the Argentinian.
He told Marca: "It is completely false that Martino has said to any of his coaching team that he is leaving.
"He is focused on the challenges ahead with Barcelona. All three titles are still to play for."
Barcelona are currently four points adrift of Primera Division leaders Real Madrid in third place, behind Atletico Madrid, and are also due to play Real in the final of the Copa Del Rey. They are also still in the hunt for Champions League glory.
Despite playing down talk of a potential departure for the 51-year-old coach, Arroyo confirmed that there is a clause in his contract to allow him to leave when the season is other if "both parties" reach an agreement.
Martino was confirmed as Barcelona manager on July 22 last year and has previously managed Argentine side Newell's Old Boys and Paraguay's national team.