Griezmann is Barcelona’s top target for the summer after the capture of Philippe Coutinho in January.
And the La Liga leaders are expected to launch a bid for the France ace when the season finishes.
Spanish outlet Don Balon claim the move is 90% done and he is already looking for places to live in the city.
Barca will pay Griezmann’s £88m release clause and are expected to beat Manchester United and Real Madrid to his signature.
United had been expected to sign Griezmann in the summer before the signing of Romelu Lukaku.
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And Alexis Sanchez’s arrival at Old Trafford means Jose Mourinho no longer wants any new forwards in the coming summer.
"The objective is that now we have more number nines," he said.
"We have Alexis, we have (Romelu) Lukaku, we have (Anthony) Martial, we have (Marcus) Rashford.
"I don't want attacking players, so don't speak about attacking players that are coming here, because nobody is coming here."
Real Madrid meanwhile are focussing on the signing of Neymar as they look to overhaul their attack.
A move for Neymar would be complicated, although not impossible, because of their cross-town rivalry making Barca favourites.
Ousmane Dembele, who joined Barcelona in the summer, is reported to know that his compatriot is on his way.
But Barca may have to sell to buy with the transfer budget being revealed.
Vice-president Jordi Mestra said: “Each season we have the same amount: £53m plus sales.
“The wage bill is high, yes, because having the players we have is not cheap. Our priority is sports excellence and economic rigour. One endures the other. They are very linked.”
The Neymar sale last summer helped finance the signings of Dembele and Coutinho but a number of fringe players are expected to leave the club in the summer.
