Manchester United have long been determined to sign Griezmann, who scored 26 goals from 53 games for Atletico Madrid this season.
At the start of the week it was claimed that the France international had told the Spanish giants he intended to quit and move to the Premier League.
However, in the wake of the club's failure to have their transfer ban overturned, the deal now looks to be on hold.
Atletico are unable to sign a replacement if Griezmann did leave, plunging their bid to re-sign Chelsea star Diego Costa into doubt as well.
United, meanwhile, have looked at alternatives amid news that the 25-year-old could now extend his stay in the Spanish capital.
Manchester United will sign Antoine Griezmann next summer – Graham Hunter Star players Manchester United could target instead of Antoine Griezmann Fri, June 2, 2017
Who could Manchester United sign instead of Antoine Griezmann
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In a year's time I expect him to end up Manchester United
Graham Hunter on Antoine Griezmann
But Hunter reckons the 26-year-old will eventually move to United – but 12 months down the line.
"Fans would tear the old and new down if Griezmann had wanted Manchester United to pay his buyout clause or if he had said I'm not playing," he told BBC Radio Five Live.
"He's not like that, he's staying and that's that.
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"But in a year's time I expect him to end up Manchester United.
Manchester United have now turned their attentions to Everton hitman Romelu Lukaku
"He had it in his mind, and this is important, that he thought the moment was right to show his talents elsewhere."
Express Sport understands the Red Devils have now made Everton hitman Romelu Lukaku their No1 target.
And moves for Torino's Andrea Belotti, Real Madrid's Alvaro Morata and Lyon's Alexandre Lacazette have also been mooted.
Griezmann, meanwhile, scored as France thumped Paraguay 5-0 in Paris last night.
Arsenal's Olivier Giroud netted a treble with Griezmann getting on the scoresheet along with Tottenham flop Moussa Sissoko.