Reyes, who is set to sign a four-year deal, will have a medical on Tuesday before being officially unveiled.
Atletico were seen as favourites to sign the 23-year-old for a reported £6m after Real Madrid apparently refused to offer more than £4m.
Reyes was on loan with Real last season and scored the vital goals against Mallorca to help the side to La Liga.
An Arsenal statement said: "Everyone at Arsenal thanks Jose Antonio for his contribution to the club and wishes him the best of luck for the future."