Last season the 21-year-old burst on to the scene at the Camp Nou and impressed in his few appearances with the Blaugrana, particularly after La Liga recommenced after lockdown.
However, he has been a victim of a summer of turmoil at Barca.
Following the damaging loss in the Champions League versus Bayern, head coach Quique Setien was replaced by former player, Koeman, who was charged with undertaking a squad overhaul.
Various players such as Arturo Vidal, Ivan Rakitic, and Luis Suarez have been deemed surplus to requirements in Catalunya.
But, surprisingly, Puig has also made the list of unwanted stars despite coming from La Masia with an extremely promising reputation.
According to RAC1, Koeman informed Puig to find another club after he left him out of his 23-man squad in Barca's final pre-season friendly against Getafe on Saturday.
Koeman is believed to favour a two-man midfield of Frenkie de Jong and Miralem Pjanic, while he has asked the club to sign Gini Wijnaldum from Liverpool.
The Dutchman is of the opinion that Puig will not get enough game time this season due to his position in the midfield pecking order.