Atleti maintain that Griezmann and Barcelona reached an agreement before the player's much talked about buyout clause dropped from €200m to €120m in July.
Their argument is that Barcelona should have paid €200m. But the Catalans only paid €120m to break Griezmann from his contract in Madrid as that was what was stipulated after 1 July.
According to SPORT, FIFA cannot intervene because the dispute is between two clubs from the same country, which left Atletico to raise the issue of their missing €80m at a committee meeting between a handful of La Liga clubs on Thursday.
The report notes that the unsavoury situation was also then on the agenda of an RFEF meeting later the same day, although Spain's governing body is yet to reach any conclusion.
If RFEF is to intervene, SPORT explain that the matter can either be addressed by the federation's own board of directors - of which both Barcelona and Atletico presidents are members - or by referring the matter to a separate committee within the governing body to resolve the matter.
"The amount deposited is insufficient to pay his release clause because it is obvious that the agreement between the club and Barcelona was done before his fee dropped from €200 to €120 million," the statement read.
"Atletico Madrid understand that the contract was broken before the end of last season, on the basis of the facts, actions and statements of the player and the club has therefore started the proceedings it considers appropriate to defend its legitimate rights and interests."
"Atleti are disgusted by the behaviour of Barcelona and the player and particularly Barcelona, for inducing a player to break their contractual relationship with Atletico Madrid at a time in the season when the club was playing, not just a Champions League tie against Juventus, but also the league title against Barcelona itself."