But Manchester United are now ready to elbow their way into contention and try and tempt the France star to snub the Nou Camp for Old Trafford instead.
Barcelona have been in the box seat for moths, but their failure to wrap up the deal has given their rivals hope.
And United are set to bid £95m for the Atletico man, which would break their transfer record set when buying Paul Pogba from Juventus for £89m in 2016.
The Sun report the Manchester giants will offer a huge deal worth £450,000-per-week to sign for Ole Gunnar Solskjaer.
Paris St-Germain are another club interested in Griezmann, and their ability to offer the player Champions League football could give them the edge.
Griezmann has been close to joining United before.
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“Man Utd will offer Antoine Griezmann a staggering £450,000-a-week to join from Atletico Madrid”
He appeared on the brink of a switch in 2017, but a deal fell apart when he stayed at Atletico due to the club being hit with a transfer ban.
He also performed a U-turn on a switch to Barcelona 12 months ago having looked destined for a move to the Spanish giants.
Griezmann revealed he’s decided to leave Atletico after talks with top brass and manager Diego Simeone.
“After speaking with Cholo (club coach Diego Simeone), then with Miguel Angel and the people in the hierarchy of the club I wanted to speak with you, the fans, who have always given me a lot of love,” said Griezmann.
”I wanted to tell you I have taken the decision to leave.
“The truth is it has been difficult to take this route but it is what I feel I need and I would like to thank all of you for the love you have shown me during these five years.”
Griezmann is big pals with Paul Pogba – and his potential arrival in Manchester could sway the midfielder into ignoring a Real Madrid move.
“Paul is a great guy [but] because of his over-exposure in the media, his transfer fee and his spectacular style of play, everything he does is scrutinised,” he said.
“The press is very demanding with him, sometimes too harsh in my opinion.