The 31-year-old joined Atletico Madrid from Barcelona this summer as he brought to an end his three-year stay at the Nou Camp.
Balague told Revista that Arsenal tried to sign the player in January but failed to follow up that interest when Barcelona were eager to offload the forward this summer.
Villa was largely restricted to appearances off the bench in his final season at the Nou Camp as he made his way back from a broken leg, which kept him out of the game for eight months.
And this, Balague suspects, was what deterred Arsene Wenger from making another bid for the player.
"In January Arsenal came and offered a four-year deal, about 20m euros, and some money for Barcelona. Barcelona said they were not selling," Balague told Revista.
"So I think that David Villa, who took his kids to a British school, expected Arsenal to come back but they didn't, because they saw that David Villa in the last six months was not happy and wasn't playing well, and they probably thought he is not what he was.
"Atletico Madrid came in and they got a deal for 2.5m euros - that is all for the first year. If he stays for three years it is 4.6m euros. Why so cheap? I found out that Barcelona have a first option on [18-year-old Atletico midfielder] Oliver Torres.
"Now if Atletico fans find out about that they might not be happy but it is one way of Barcelona saying they are getting rid of a big wage and getting a first option on this guy, and let go someone who was not happy either."
Villa, who has replaced successful striker Radamel Falcao at Atletico, started their 3-1 victory over Sevilla and Terry Gibson hopes he can regain his best goal-scoring form this season.
"If you are going to try to get a replacement for Falcao - and it goes without saying his success at Atletico Madrid - David Villa is as good as you are going to get.
"I think they got him for a good price. I think he needs dusting down, get his sharpness back and become used to playing again in that position that he started in the other night - in the middle of attack.
"That is the old David Villa who played for Valencia and Zaragoza, and that we want to see. At Barcelona he played wider on the left. I am hoping he really gets back to enjoying his football and playing in his favourite position again."