Atletico's number 10 shirt was last worn by Belgian winger Yannick Carrasco prior to his shock move to Chinese Super League club Dalian Yifang in February of this year.
Lemar has previously been a serious target for Liverpool and almost joined Arsenal last summer until pulling the plug on a proposed £90m switch. The 22-year-old has also been linked with Barcelona this month as a cheaper option than Tottenham playmaker Christian Eriksen.
But, according to Spanish radio show El Larguero, it seems that Europa League winners Atletico have won the race for his signature.
Marca expect the deal to surpass Atletico's current club record, which stands at €55m plus add-ons to bring Diego Costa back to the club from Chelsea last summer.
Lemar emerged as a top talent during the 2016/17 season. He contributed 31 combined goals and assists in all competitions as an exciting Monaco side won Ligue 1 for the first time since 2000 and reached only their second ever Champions League.
The player wasn't as effective in 2017/18 after injury disrupted his campaign in November, although he still managed 10 assists 30 Ligue 1 appearances.
Lemar will soon embark on the first World Cup of his career with France, whose opening game will take place against Australia in the city of Kazan on 16th June.