Griezmann, who was playing at Real Sociedad four years ago, was the subject of interest from the Gunners after the Frenchman's then-agent, Eric Olhats, had been in contact with Arsenal scout Gilles Grimandi.
The move failed to materialise, however, as Griezmann joined current club Atletico in 2014, and the forward has said how he snubbed Arsenal when future interest was registered in him.
"I waited, I waited and I kept waiting," Griezmann wrote in his new autobiography, Behind the Smile. "When there was no news, Eric called Grimandi, who said that the manager was still interested in me [and] to keep waiting.
"Finally a few hours before the market closed, he let us know Arsenal would not make a move. I don't like to be told something and for it not to happen.
"So when Eric told me later that the London club were interested again, I told him 'Forget it, after the blow they gave us'".
Griezmann scored twice in his side's 5-0 win over Levante last weekend.