Lemar is now away on global 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.
Last weekend, Atletico Madrid forward Griezmann said he would confirm his plans before France begin their World Cup campaign against Australia on Saturday.
After a great deal of uncertainty surrounding Lemar's next club, reports have emerged suggesting the French winger will play his football at the Wanda Metropolitano next season, ostensibly linking up with national teammate Antoine Griezmann as Los Rojiblancos look to challenge the likes of Barcelona and Real Madrid for Spain's top honours.
Catalunya Radio reported that Barca remain confident of signing Griezmann as coach Ernesto Valverde was reassured by the club's hierarchy over a meal last week that the forward would be part of his squad next season.
"Today is not the day to talk about my future", he said at a press conference on Tuesday.
Monaco have already sold Fabinho to Liverpool this month, and they are set to make further profit with the sale of Lemar.
Griezmann revealed today he had decided where he will be playing next season but claimed it was not the right time or place to announce it.