Griezmann, 26, remained at Atletico Madrid last summer out of loyalty to the club.
He admitted he was “six out of 10” likely to join Manchester United but Atletico’s transfer ban meant they couldn’t sign a replacement striker, so he remained and signed a new contract.
However, Griezmann had an £88million (€100m) release clause inserted in that contract, which will be active next summer.
And Barcelona are preparing to trigger that clause.
That is according to Spanish newspaper Mundo Deportivo, who claim Barcelona are aware of rival interest in the striker.
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Manchester United and Real Madrid also supposedly want Antoine Griezmann
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Antoine Griezmann (right) will likely play for France at the World Cup in Russia next summer
At £88m Griezmann would be deemed a bargain signing, with transfer fees passing the £100m mark for the first time last summer.
It is claimed manager Ernesto Valverde is a massive fan of the France international and will back him in the transfer market.
Relations at the club were strained in the summer when Neymar was forcibly sold to PSG for £200m and Barca struggled to reinvest the money.
Philippe Coutinho is another transfer target but Liverpool have proven difficult to negotiate with.
Jesse Lingard reveals he wants Antoine Griezmann at Man United
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Unlike Griezmann, Coutinho doesn’t have a release clause.
Mundo Deportivo claim Barcelona want to secure Griezmann before the World Cup in Russia starts in June next year.
They hope that before than a verbal agreement with Griezmann’s representatives can be made.
Griezmann has scored just three goals for Atletico this season - his last strike coming in the 2-1 Champions League defeat to Chelsea back in September.