Griezmann is contracted at Atletico - the club he joined in 2014 - until 2022, but Barcelona are reportedly prepared to trigger his €100million release clause.
Griezmann is expected to announce his future before his nation's World Cup campaign begins against Australia on Saturday.
This (the World Cup ) starts on Saturday, so it will be defined this week.
He was just your typical football mad seven-year old in awe of his idols in the flesh at the time, but Griezmann was already in his full kit, wearing the No.7 shirt - a number which he has worn with distinction ever since becoming France's main man.
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"It's good that you're taking an interest", Griezmann said after France's 1-1 friendly draw with the United States.
Many saw his press conference on Tuesday, four days before France's first fixture of the competition, as the ideal chance to do just that.
But he told a news conference: "I apologise - I am really very sorry".
"I know there's a lot of expectation, I know many were waiting for it, but today is not the day (to speak about my future)". However, Atletico responded with a counter-offer which will make him the club's best paid player by some distance.
After facing the US on Saturday in Lyon, France are due to fly to their training base near Moscow on Sunday. And Griezmann is one of them.