The north London side have been relatively busy in the summer transfer window so far, and the north London side on Friday completed the permanent signing of Martin Odegaard from Real Madrid.
Odegaard arrives at The Emirates after his six-month loan spell with the Gunners last season, and he follows the likes of Albert Sambi Lokonga and Ben White to the north London side.
Arsenal have also been credited with an interest in FC Barcelona star Coutinho in recent days, although that was before the Gunners completed the signing of Odegaard.
Former Liverpool FC and England winger Barnes believes that Coutinho would be a great fit for Arsenal and feels that the Gunners should try and get a deal done if they can.
Speaking in an interview quoted by Metro, Barnes said: “A move to Arsenal would be a good one for Philippe Coutinho, if the possibility arises.