Griezmann has revealed his intentions to leave Atletico Madrid after five years but the club are yet to receive any official bids.
Barcelona were expected to swoop for the Frenchman but have so far failed to firm up their interest with an offer.
However, Mundo Deportivo claim he is confident a move will materialise once his release clause drops to £105million on July 1.
Manchester United have also been heavily linked and this latest update will come as a blow to their hopes of pulling off a surprise move.
He even hinted he might look to leave Spain after starring for France in their 4-0 win over Andorra.
“There’s a need for patience, as it’s still soon,” Griezmann said.
“I want my future to be decided more than anyone, but there’s a need to wait.
“I don’t know if I’ll remain in Spain. Maybe in two weeks we’ll know something.
“I just want to play football and have fun.”
It had been claimed United might look to steal him from under Barca’s noses as they could be struggling to secure a deal.
They have already paid Ajax £60m to sign Dutch sensation Frenkie de Jong and are one of the frontrunners in the race to sign his fellow Dutchman Matthijs de Ligt.
De Ligt could cost up to £80m, although Paris Saint-Germain are now the favourites to and him.
Whatever happens with Griezmann, United boss Ole Gunnar Solskajer is desperate for reinforcements after a disappointing end to last season.
The Norwegian has already signed winger Daniel James from Swansea and is looking at another 12 potential targets.
A new centre-back and a holding midfielder remains a priority and Express Sport understands West Ham star Declan Rice is one of his top targets