It was reported last year that Arsene Wenger tried to sign the Spaniard but neither club has ever confirmed that any bid was lodged.
However, Wilson told talkSPORT: "Arsene tried like mad to buy not only a goalkeeper but what he considered to be the most consistent in the country, so he made a big bid for Pepe Reina.
"That didn't come out because that's not Arsene Wenger's way of doing it and it's not Arsenal's way of doing it, but he did attempt to do that."
The performances of Wojciech Szczesny this season have eased Arsenal’s goalkeeping worries somewhat, although his injury has forced Wenger to bring Jens Lehmann out of retirement to provide cover for the rest of the season.
But Wilson, who coached the club's goalkeepers for almost 30 years after playing for the Gunners, believes that Wenger would have let the promising Pole leave had he managed to capture Reina last year.
"I think he would have lost, without any doubt, the young kid Szczesny, who I believe is going to be the future," Wilson added.
"It's most unfortunate he has had this injury and in the game that he had it, but I happen to think Szczesny is going to be a great goalkeeper. The boy is ready now - you ask Arsenal fans."
And Wilson believes Lehmann will be a useful addition to the Gunners squad, even if he doesn't play that much.
"Jens would be incredible in the dressing room and I do believe he is fit enough to play," he added.
"For Jens to be within the Arsenal set-up at this crucial run-in time I think it could be terrific. I think he would very quickly get into his shot-stopping, you just do not lose that. You only have to look at Edwin van der Sar.
"He looked no different from when he played with us and we know he's a conscientious animal. And my goodness me what an input he would have in the dressing room, having been part of the Invincibles team."