The 22-year-old is now at the World Cup with France, who are due to play their first game of the tournament on Saturday against Australia.
France kick off their World Cup campaign against Australia on Saturday and Griezmann had previously insisted that he would reveal his future before that match.
"It's good that you're taking an interest", Griezmann said after France's 1-1 friendly draw with the United States.
But he told a news conference: "I apologise - I am really very sorry".
Monaco have already sold Fabinho to Liverpool this month, and they are set to make further profit with the sale of Lemar. I have made my choice.
But those closest to the 27-year-old are urging him to stay at Atletico, sources said, with one source indicating there is no sign at this time that Griezmann is planning on a move to Barcelona.
Puig officially extended his contract with Barcelona and now has a release clause of €100 million, which should fend off any potential suitors for a while.
"He is a complete player but it is his decision".
Lemar scored three goals and provided 10 assists this past season.
Dermot Corrigan is a Madrid-based football writer who covers La Liga and the Spain national team for ESPN FC.