With the addition of Miralem Pjanic and Carles Alena impressing in preseason, Koeman believes the midfield is overstocked.
And he has informed La Masia graduate Puig, 21, that there is no space for him in the squad.
Puig, who made 12 appearances last season, played no part in Barca’s Joan Gamper Trophy win over Elche on Saturday.
The Spain under-21 international was one of the positives in a disappointing 2019-20 season for Barca.
There is already talk that Ajax could make an offer for Puig, potentially in a deal for Barca target Sergino Dest.
Monchu to Girona
21-year-old Monchu is one of Barca B’s best players and a string of clubs have shown interest in him.
Girona asked for Monchu on loan several days ago and there is hope that a deal can be finalised this week.
The second division club have already signed Manchester City talents Aro Muric and Pablo Moreno on loan deals.