Having identified the France global striker as a top target, those at Camp Nou have set about trying to put a big-money deal in place.
Speaking ahead of next week's final, Gabi told a press conference: "We are very relaxed, focused on the final, then we'll see afterwards about Griezmann's future".
"Griezmann is a great player, there is no doubt about that, but we have to be careful and respectful when we're talking about players that don't play for us", he told reporters, via Sport.
"I also told him in that moment, that in the chance the player exercised his right to nullify his contract due to their pressure throughout the season that Atletico Madrid would demand Barcelona compensation for their inappropriate behaviour".
While a stunning quarter-final collapse to Roma cost them a shot at Champions League glory, Bartomeu believes Messi does not have to leave to build on his four European titles.
He finished off by declaring "enough is enough".
Officer on way home from work saves life of unresponsive baby
This is a representational image showing Los Angeles County Sheriff's deputies in Compton, California, Jan. 29, 2015. His mother commented to say she was extremely grateful for Deputy Nix's help and provided more details on the rescue.
Since they are expected to make in a swoop for numerous players including Griezmann this season, Barcelona will be looking to puff up their purse even more in the next couple of months.
"Enough is enough. I hope this statement serves its objective and that us Atletico fans can enjoy this moment and prepare ourselves for a final like we should", the statement concluded.
"We've talked with Cerezo and Gil Marin about many things".
Griezmann, one of Atletico's key players, has been heavily linked with a move to the Nou Camp at the end of the season, with Spanish media reporting that Barca are willing to pay his 100 million euro (87.57 million pounds) release clause. "[Cerezo] is the president of the club he plays for".
But now Costa has urged Griezmann, who was wanted by Man Utd last summer, to think hard before jumping ship to Barcelona.
Griezmann has a release clause of €100m, and given the relative straightforward nature of a signing, Ernesto Valverde's side have "taken for granted" the 27-year-old's arrival.