The France international, who is under contract with the capital club until 2021-22, has been linked to moves to moves away from the capital over the past year, with Manchester United among the teams mentioned.
However, reports of Barca interest have grown in recent weeks, with star Barcelona striker Luis Suarez welcoming the idea of a Griezmann move to the club .
There have also been hints from Barcelona's leadership that they have at least met with people close to Griezmann - last week Barcelona director Guillermo Amor admitted the possibility that club president Josep Maria Bartomeu may have had contact with people close to the forward.
"It could be that Bartomeu had a meeting with Griezmann's family," Armor told Movistar.
"If that's what the newspaper says, it may be that way, although I do not know.
"There is good harmony, a good relationship, there can be conversations, rumours are inevitable in the world of football."
Atletico believes these communications violate La Liga rules as Griezmann is under a long-term contract with the club, and such contact is only permitted in the last six months of a player's deal.
According to Atletico, this violation is made worse by the position of the two clubs, as they are second in the table, six points back of Barcelona, and could have impact on the title race.
Should Barcelona be found guilty of the violations, a punishment severe as a two-window transfer ban may be handed out.