AS and Marca claims that Atletico have decided to present a formal complaint to Fifa after it emerged that Barcelona official had met with the Frenchman's family to sound out his potential move to the Nou Camp.
The 26-year-old only signed a new lucrative deal with Diego Simeone's side in June following speculation linking him with a move to Manchester United.
It was said that Griezmann only put pen to paper on a new deal because Atletico were unable to sign a replacement in the summer after the Court of Arbitration for Sport (Cas) upheld their Fifa transfer embargo.
The rumours linking him with both United and Barcelona have therefore continued during an opening part of the season in which Griezmann has sent contradictory messages regarding his long-term future at Atletico.
In October, Mundo Deportivo reported that manager Ernesto Valverde had identified the Atletico star as his number one target to bolster the attack ahead of the 2018-2019 campaign.
Mundo Deportivo fuelled speculation over the weekend after revealing that Barcelona president Josep Maria Bartomeu recently met with the Frenchman's family to discuss the proposed move.
The Spanish publication added that Barcelona would be ready to meet his full €100m (£88.5m, $118m) release clause to beat United for his services and lure him to Valverde's side in the summer.
Barcelona head of sporting and institutional relations Guillermo Amor did little to deny the meeting when asked about those reports following the Catalans' 4-0 victory over Deportivo La Coruna.
"Well. If the newspaper says so, it may be the case. There may well have been an approach," Amor admitted to Movistar. "I don't know. There is a good relationship, there can also be conversations. Rumours are inevitable in the football world. In Griezmann's case there are many players that you can talk to and enter some kind of relationship with, in the event an opportunity arises. The club usually carry out this kind of work."
Marca says that Atletico believes that Barcelona's tactics are against the integrity of La Liga as it destabilize Simeone's side.
Los Colchoneros have emerged as the main candidates to beat Barcelona for the title this season, and currently occupy second place in the table, six points behind the Catalans.
Gremio recently threatened Barcelona with legal action after a photograph of Brazilian starlet Arthur Melo wearing a Blaugrana shirt alongside club technical secretary Robert Fernandez was leaked through social media.