Spanish newspaper Mundo Deportivo says the Catalans will pay Griezmann's release clause of 120m euros ($135m) and the 28-year-old has signed a five-year deal at the Camp Nou.
French daily L'Equipe add Barca have already sent the bank transfer to the capital city club.
Griezmann failed to show up for his side's pre-season gathering on Sunday after he had announced in May he would be leaving the Wanda Metropolitano but could not reveal his destination until his release clause dropped from 200 million euros to 120 million euros on July 1.
On Friday, Atletico accused Barcelona and Griezmann of a "lack of respect" after the Barca president Josep Maria Bartomeu revealed the two clubs had held talks about the former Real Sociedad forward.
Griezmann joined Atletico in 2014 and has scored 133 goals in 257 appearances lifting the Copa del Rey as well as the Europa League.