Lemar is now away on worldwide duty with France and both clubs will be keen to finalise the transfer before the start of the World Cup, which gets underway on Thursday.
Some heartwarming footage has emerged of a young Antoine Griezmann meeting his France heroes back in 1998.
Antoine Griezmann intends to clear up his future before the World Cup begins, saying after France's draw with the United States on Saturday that he will reveal all this week.
Griezmann is now valued at £200m, but will see his Atleti release clause half in price at the start of next month. "The decision is complicated, I am calm", he said. I have made my choice.
The turning point has been the differences within his family over the final decision, with some keen for him to remain in Madrid and others backing a move to Barcelona.More news: Spain offers to take migrants stranded in Mediterranean Sea
"It's not the time or the place".
France play their opening game of the tournament on Saturday against Australia.
He has scored three goals in 12 matches with the national team since his November 2016 debut.
Barca's pursuit has angered Atletico, who in December reported them to Federation Internationale de Football Association for making an alleged illegal approach and followed up with a statement in May that said "enough is enough".
"Atletico de Madrid and AS Monaco have achieved a preliminary agreement over the transfer of Thomas Lemar to our club", for the definitive transfer", the Spanish club said of the 22-year old.