Cadena Ser show El Larguero claims that the France international remains a priority target for Ernesto Valverde and the La Liga giants will make every effort to sign him ahead of the 2018-2019 season.
The Spanish radio station says that Barcelona are ready to offer Griezmann a staggering five-year deal which will see the Frenchman earning as much as Luis Suarez and only less than Lionel Messi at the Nou Camp.
Cadena Ser says that the negotiations between Barcelona and the Frenchman are already "very advanced" but they add that Atletico plan to match any offer from the Catalans to convince Griezmann to remain at Diego Simeone's side.
However, the France international instead decided to commit his future to Simeone's side by signing a new long-term contract after the Court of Arbitration for Sport (Cas) upheld their Fifa transfer embargo.
Cadena Ser then claimed that Griezmann became the best player of Atletico as a result, with Los Colchoneros increasing his wages to around €12m-year (£10.6m, $14.3m) including add-ons - equivalent to €230,000-a-week.
But speculation regarding a potential move away from Atletico have continued with reports claiming that both Barcelona and United are still keen on his services ahead of the summer transfer window.
Indeed Last month Atletico reported Barcelona to Fifa after it was reported that club president Josep Maria Bartomeu had met with the Frenchman's family to sound out his potential move to the Nou Camp.
But nor the threat from Fifa nor the arrival of Coutinho are expected to change the Catalan's plans over Griezmann with Cadena Ser claiming that they plan to break the bank again to sign the Frenchman in the summer.
Barcelona would pay his €100m (£88.1m) release clause at Atletico but will also give Griezmann a considerable pay rise to convince the Frenchman to complete the transfer.
Cadena Ser doesn't disclose the specific amount but they say that Griezmann will be in the same bracket as Suarez so it will surely mean a considerable increase to his current €12m-a-year wages.
Atletico won't be able to prevent his departure if Griezmann wants to leave and Barcelona trigger his €100m release clause.
Cadena Ser claims that Los Colchoneros won't give up so easily and will match the offer from the Catalans with the hopes Simeone can enjoy a fearsome attack with Diego Costa and him at least one more season.
Meanwhile, Barcelona may also need to battle with Manchester United to his signing with Jose Mourinho said to be still monitoring the situation.
It has been said that Griezmann would rather to stay in La Liga but earlier this week The Sun claimed that the player had demanded from United a staggering £400,000-a-week deal to snub the Catalans for a move to Old Trafford.