The Gunners were interested in three Barcelona players - Messi, Gerard Pique and Cesc Fabregas - before they broke into the first team at Camp Nou, but were forced to settle for only the latter, who went on to make more than 300 appearances for the club.
Pique instead joined Manchester United before returning to Barcelona, whereas Messi stayed at Barca his entire career and has since established himself as arguably the greatest footballer of all time.
"The story of Cesc Fabregas is that Franny Cagiago and Steve Rowley brought the player here and we had to convince him. I met Cesc's parents and at the time we were interested in Messi and Pique as well," Wenger told Arsenal's official website.
"We tried for the three but of course it didn't work out, but we got a gem there in Cesc and he is an exceptional player. A brain for football. [It didn't work out with Messi and Pique] because of the agents. I think it was linked with Nike at the time and they wanted Pique to go to Man United.
"With Messi, Barcelona didn't want to lose him of course and they made [the offer] that was needed to keep the player at the club. I don't really know if Messi was interested... I couldn't get close to try to force the deal because Barcelona stopped that possibility very early."
Wenger stepped down from his role as Arsenal manager after 22 years at the helm this summer.