Lemar, 22, had been linked with moves to Liverpool and Arsenal over the past 12 months but now he looks set to make the switch to La Liga.
Atletico said in a statement on their official website on Tuesday: "Atletico de Madrid and AS Monaco have achieved a preliminary agreement over the transfer of Thomas Lemar to our club.
"Over the next days, both clubs will work to close the agreement for the definitive transfer."
The France international, currently in Russia with Les Bleus for the World Cup, joined Monaco in 2015 from Caen and made his international debut in November 2016.
Monaco echoed Atleti's statement on their official Twitter feed, saying: "AS Monaco and @Atleti announce that they have found an agreement in principle for the transfer of Thomas Lemar.
"The two clubs will continue to work in the coming days in order to find a definitive agreement."
The move came on the same day as fellow France forward Griezmann said he has decided on his future - but has refused to say whether he will leave Atletico this summer.
That led to Griezmann being cut to 11/10 to join Barcelona this summer, as he's been linked with the Spanish champions for some time.
Griezmann and Lamar will line up for France in Group C of the World Cup, where they will face Australia, Peru and Denmark.
