Antoine Griezmann must choose if he wants to make history at Atletico Madrid or "go to a club where he will never make history,'' Los Rojiblancos' chief executive Miguel Angel Gil Marin has said.
Griezmann, who netted a brace as Atleti beat Marseille 3-0 in Wednesday's Europa League final in Lyon, has been strongly linked with Barcelona.
The France forward was quoted by Marca as saying after the game: "Now is not the time to talk about the future.''
Atleti boss Diego Simeone, meanwhile, said the club would continue to grow whether they lost Griezmann or not.
Gil Marin also spoke on the matter, being quoted by Cadena SER as saying: "It's down to his feelings, not those of his family or his entourage.
"It's him who must decide whether he wants to make history at Atletico Madrid where everything revolves around him, or go to a club where he will never make history.
"He is human and we must respect his decision.''
Gil Marin last week said in a statement regarding Griezmann published on Atletico's official website that they were "fed up with Barcelona's attitude,'' and accused the Nou Camp outfit of showing "an absolute lack of respect'' towards Atleti with recent comments about the player.
That was shortly after Barca president Josep Maria Bartomeu revealed he met with Griezmann's agent earlier in the season.
Griezmann, who has been with Atletico since 2014, is on a contract that runs to 2022.
On Wednesday, the 27-year-old struck in the 21st minute and then again four minutes after the interval, before an effort from skipper Gabi late on wrapped things up.
With the Europa League clinched, Simeone's side will next look to seal their current position of second in LaLiga when they host Eibar on Sunday in their final game of the campaign.
They are three points clear of third-placed Real Madrid, who have a significantly better goal difference. Real are at Villarreal on Saturday.