The Real Madrid star was seen as the perfect replacement for Alexis Sanchez who is refusing to sign a new deal at the Emirates.
Spanish outlet Don Balon claim the Gunners were determined to sign Bale this summer.
But the Wales international has no intention of leaving the Bernabeu for Arsenal although he is keen on a Premier League return.
Bale is said to have snubbed Arsene Wenger because his side will not be playing Champions League football this season.
The winger is concerned about his future at Real with the emergence of Marco Asensio and the expected arrival of Kylian Mbappe.
But Bale has made his mind up and will not be heading to Arsenal this summer.
The other Premier League side linked with the 28-year-old are Manchester United as Jose Mourinho looks to overhaul his squad.
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The Special One was filmed talking to Bale after United’s pre-season win over the Spanish side.
He told him: “I can't buy you because you don't talk to me.”
Arsenal will be desperate to sign a replacement for Sanchez if the Chile international does indeed leave the club.
The likes of Manchester City, Bayern Munich and Juventus have all been linked with the forward.
