"Over the next days, both clubs will work to close the agreement", the Madrid club said of the attacking midfielder usually deployed on the left.
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Having previously stated that he would announce his decision before France's opening World Cup game on Saturday, Griezmann chose to keep quiet.
"We understand there won't be easy matches at the World Cup for us", the Atletico Madrid star told French sports daily L'Equipe.More news: Kim Jong Un goes for surprise nighttime walk ahead of summit
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Messi is excited by the prospect of uniting with the 27-year-old and believes he would go a long way to helping Barca become European champions for the first time since 2015.
Griezmann signed a one-year contract extension with Atletico last summer that runs until 2022.
Lemar and Griezmann are a part of the France team that will begin their Group C campaign against Australia on Saturday, before facing Peru and Denmark in the group stage.
He made his France debut in 2016 and has scored three goals in 12 worldwide games.