France forward Griezmann seemed destined for Camp Nou when he declared the 2018-19 season would be his last with Atletico.
Barca reportedly came close to signing the World Cup winner 12 months ago and were expected to make a fresh approach following their Champions League failure.
The Catalan club are now rumoured to have cooled their interest and Atletico president Enrique Cerezo admitted to being entirely unclear on his star player's next destination.
"Griezmann has been with for five years and done a splendid job," Cerezo told Spanish broadcaster Onda Cero.
"We have a good relationship. Nice, trustworthy.
"I hope wherever he goes he will be treated the same as he has been with us.
"I don't know where he's going, he didn't say. I don't have the slightest idea where he will go. I don't think anyone knows."
Asked whether Barca have initiated discussions, Cerezo replied: "No one from Barca has contacted me."
Griezmann joined Atletico from Real Sociedad in 2014.
The 28-year-old is with France ahead of their Euro 2020 qualifiers against Turkey and Andorra.