Arsenal allegedly failed with a bid to sign Barcelona midfielder on loan during the January transfer window.
Having finally been able to part ways with Mesut Ozil, the Gunners were in a position to bring in a new creative player at the midway point of the campaign.
While Arsenal eventually wrapped up a deal for Real Madrid playmaker Martin Odegaard, it appears that they also moved for another prospect playing in La Liga.
According to Italian journalist Fabrizio Romano, Arsenal wanted to sign Puig, initially on loan with the option to buy on a permanent basis in the summer.
Barcelona made the decision to retain the services of the 21-year-old, who recently agreed to pen fresh terms at Camp Nou.
Puig has made nine appearances for Barcelona since the turn of the year after proving himself to boss Ronald Koeman.