The details are outlined in a report from El Mundo Deportivo, who say the Dutchman is a popular figure with the first-team players and would be a popular successor to the under-fire Setien.
The former striker netted 122 goals in 257 games for the Catalan giants and he is currently the director of youth football at the club.
Kluivert was appointed in the role last summer having previously been the number two to his former Dutch international teammate Clarence Seedorf at the helm of the Cameroon international side.
He is said to be regarded in a similar mould to Zinedine Zidane at Real Madrid as he demands authority due to his playing career whilst also being popular and knowing how to handle a dressing room at the elite level.
Kluivert speaks Dutch, German, English, French, Italian and Portuguese while he is said to have played a key role in attracting youth players to the club over the past year.
The Blaugrana conclude their La Liga campaign against Alaves on Sunday before hosting Napoli in the second leg of their Round of 16 Champions League tie next month.
It has been suggested Setien may be removed from his position before the European tie and Kluivert could be an ideal continuity replacement.