The 22-year-old Monaco winger was on the verge of crossing the English Channel last year when Arsenal and Liverpool made high powered bids in the final hours of the summer transfer window.
The 22-year-old is now at the World Cup with France, who are due to play their first game of the tournament on Saturday against Australia.
"Over the next days, both clubs will work to close the agreement", the Madrid club said of the attacking midfielder usually deployed on the left.
Griezmann's reluctance to reveal his plans could be because he is playing a waiting game to make sure he gets the best deal possible to renew his contract with Atletico.
Atletico Madrid have announced an agreement has been reached to sign Monaco star Thomas Lemar for a reported fee of €60 million (£52.9m/$70.5m).
With ongoing speculation linking him with Barcelona, Griezmann had said he would confirm his position before France's World Cup opener against Australia on Saturday.
After announcing a press conference for Tuesday afternoon, it seemed that he was set to announce where he will play next season, only to opt against it.
The duo are being interviewed at the same time, reeling off names of their favourite players as the presenter says: 'They are so cute we can not refuse them anything'. Griezmann had been strongly linked with a move to Barcelona.
Lemar, 22, has become a regular fixture in the France national team squad since making his worldwide debut in 2016.
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