Griezmann is the third high-profile player to leave Atletico this summer after defender Lucas Hernandez joined Bayern Munich and long-serving captain Diego Godin left to join Inter Milan after his contract expired.
Barcelona, which won La Liga last season by nine points, has strengthened itself at the expense of second-placed Atletico Madrid by signing its leading scorer in each of the last five campaigns.
He was scouted by Basque club Real Sociedad at the age of 14.
Atletico Madrid say they are owed a further 80-million Euro for Antoine Griezmann.
The French forward will be presented as a new Barça man on Sunday but the club wanted to release a short video in which Griezmann had a message for the supporters.
The move ends the speculation regarding the future of Griezmann, who made 257 appearances, scoring 133 goals and assisting another 50 strikes, for Los Rojiblancos.
United can nearly certainly forget about signing Griezmann now despite tracking him for the past few years.
As well as his World Cup success in Russian Federation last summer, he is also a Champions League and European Championship runner-up after Atletico and France fell at the final hurdle in those respective competitions in 2016.
Griezmann was seemingly set to join Barca in the summer of 2018 only to confirm during a television documentary he was staying at Atletico.
Barcelona said the move can help cement Griezmann's place among the world's elite performers.