Reports emerged in the press last summer that Arsenal had made a substantial bid for Reina, with The Sun claiming it was worth £23 million and the Daily Mail putting it at £25 million.
Wilson has now confirmed that Wenger - who has faced criticism for his failure to sign a top-class goalkeeper - did make an offer for the Spain international.
"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," Wilson told talkSPORT.
"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."
While an injury crisis has forced Wenger to re-sign Jens Lehmann on a rolling contract, the emergence of Wojciech Szczesny has at least provided hope that Arsenal now have a goalkeeper capable of establishing himself long-term.
Wilson believes that Wenger would have had to have allowed Szczesny to leave the club had Reina joined.
"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."