Atletico believes the LaLiga champion should have paid €200 million ($322.6 million) to meet the forward's release clause, insisting it is "owed an additional €80 million to ratify the transfer" in a statement released on Saturday (AEST).
The clause in the 28-year-old's contract dropped to €120 million on 1 July but Atletico claims Griezmann agreed terms before that date, with club president Enrique Cerezo saying he was unhappy with the manner of the France international's departure.
Cerezo said he was disappointed with Griezmann's attitude, saying: "He wanted to leave, but he didn't tell us until the last minute."
Karian told L'Equipe that the feeling is mutual: "Antoine is very disappointed by the attitude of Atletico and its leaders, who acted with mind-blowing bad faith and communicated the opposite of what they told us in private.
"Let the leaders of Atletico do as they see fit. We will do the same.
"But for the moment, for him [Griezmann], what he wants is to spend this weekend discovering his new home."
Barcelona revealed on social media on Sunday (AEST) that Griezmann had arrived in Barcelona ahead of his official presentation.