The Gunners were actively on the search for a new left-back last month and they made a late approach for Firpo.
However, their proposal was snubbed by the Catalan giants, who had no time to bring in a suitable replacement.
According to AS, the Blaugrana are likely to sanction Firpo's exit this summer as they aim to bring in Nuno Mendes.
Mendes, 18, has excelled for Sporting Lisbon this term, where he has made 25 appearances across all competitions.
Should they manage to land him, they are likely to part ways with Firpo and the Gunners will be the front-runners to sign him.
The Blaugrana are likely to demand around £20m to offload Firpo, who was previously in the ranks of Real Betis.
Firpo has made only six appearances in La Liga this term with Jordi Alba being the undisputed left-back choice.
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