The 31-year-old joined Atletico Madrid instead, and Guillem Balague told Sky Sports' Revista de la Liga: "In January Arsenal came and offered a four-year deal, about 20m euros, and some money for Barcelona. Barcelona said they were not selling.
"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."