The Gunners are keen on a new, creative midfielder to add to their midfield options, with them struggling at the middle of the park, failing to sign Lyon's Houssem Aouar and Borussia Dortmund's Julian Brandt during the summer.
However, signing Puig is seen as much more opportunistic, with The Athletic claiming that Arsenal could go in for the 21-year-old.
Arsenal have scouted the Spaniard on several occasions, and according to reports, Puig will be allowed to leave the Camp Nou in January after a falling out with Koeman.
El Pais reported that Koeman accused Puig of leaking information in front of the whole squad on the Spaniard's future at the club, after Puig was not even included in several matchday squads.
As a result, Koeman is determined to shift the 21-year-old on to another club, not wanting such harmful behaviours at the club.
Puig has been made redundant at Barcelona, making a single La Liga appearance this season, a three-minute cameo against Getafe.
Speaking on why Puig has not been given any opportunities to play, Koeman simply said "Sometimes it's not just about quality. There may be players who don't do enough in training for the coach to give opportunities."
Puig has been made available for either a loan or else a permanent transfer, and this may be a great opportunity for Arsenal to grab the Spaniard for a budget price.
However, there are also concerns over the 21-year-old's attitude, something that Arsenal do not need right now given their dire situation in the Premier League.
Additionally, Puig is reportedly happy at Barcelona, despite his lack of playing time, and that Arsenal might be fearful of another Denis Suarez situation, another Spanish midfielder that struggled immensely during a loan deal in North London.
Apart from Arsenal, Leeds United are also reportedly interested in signing Puig.