The young fullback is wanted by the Ligue Une newcomers and Barca believe a loan move for 12 months would benefit Fort's development, says Mundo Deportivo.
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The Catalan giants would prefer a straight loan arrangment to maintain control of the player's future.
Paris FC are in contact, but are facing competition from several La Liga clubs.
Fort recognises he needs to leave Barca for regular football as he sits behind Jules Kounde and Eric Garcia in the right-back pecking order.