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The Catalan giants need to reduce their wage bill and have accepted they'll need to part ways with Griezmann to achieve that.
Griezmann arrived from Atletico Madrid in a £100 million in 2019 and finding a buyer is proving to be difficult.
It means Barcelona are exploring swap deals involving the France international, with Radio Catalunya claiming City have been contacted.
Barcelona have named Riyad Mahrez, Raheem Sterling, Bernardo Silva or Aymeric Laporte as four players they'd be interested in.
Pep Guardiola isn't said to be interested in the 30-year-old however and wants to keep his squad together.
Sterling and Mahrez regularly start on the wings and Aymeric Laporte is a key defender alongside Ruben Dias and John Stones.
Bernardo Silva has been linked with a move away this summer, but remains a popular player at the Etihad Stadium.
City won the Premier League and Carabao Cup last season but lost the Champions League final to Chelsea.
They also parted company with club legend Sergio Aguero, who has ironically signed for Barcelona as a free agent.
It means Griezmann is unlikely to make a move to the Premier League.
Griezmann played under Diego Simeone at Atletico for five years and the Argentine coach still has a fondness for his former player.
He told Marca: "I have a very good relationship, a very beautiful relationship beyond sport, my kids play with his kids.
"Above all, for what he gave Atletico, I want him to do well at Barcelona, he is an extraordinary player.
"I hope that he can consolidate the player he has been at the Euros, in the World Cup, in the five years with us. I don't wish for anything else but him triumphing at Barcelona.
"We speak with the club, there are positions to improve in the team - we need a striker. Another will come if Griezmann does not. The club will decide what is economically best for the club."
Barca are yet to register Aguero, Eric Garcia and Memphis Depay due to financial issues.
Do you think Antoine Griezmann will leave Barcelona this summer?
Featured Image Credit: PA Images
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