Wenger made the comments during an interview for Arsenal.com. He coached his last game for the Gunners on Sunday. Arsenal tried to acquire the superstars in 2003. Ronaldo signed with Manchester United before his run at Real Madrid. Messi has played for FC Barcelona for each of his five Ballon d'Or awards.
"I met [Cesc Fàbregas's] parents and at the time we were interested in Messi and [Gerard] Piqué as well," Wenger said.
"We tried for the three but of course it didn't work out ... 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."
He spoke about eventually reeling in Mesut Ozil before the topic changed to Ronaldo.
"The one that comes to mind is Ronaldo," Wenger said. "He was here with his mother and we were very close. Then Man United came in and they had Carlos Queiroz at the time, who was their coach. [United] played against Sporting Lisbon and Ronaldo was outstanding and they signed him."
"So you could imagine at the time to have Thierry Henry and Ronaldo together ... it would have changed a little bit the history of my stay here as well."
"There is always something you could have done differently. The problem of negotiations is to know when you give in and when you don't give in. Once we were at £4.5m, we were still in negotiations. David [Dein] met with [the agent Jorge] Mendes in Paris and he was representing Sporting, so we were very close but Man United went to £12m, which we could not afford.
Ronaldo, 33, was 18 years old when he signed with the Red Devils. Messi was 15 years old during the summer of 2003. They now have a combined 10 Ballon d'Or awards.