The 25-year-old is also thought to be wanted by Chelsea, Bayern Munich and Paris Saint-Germain after another stellar campaign in front of goal for the Rojiblancos.
Griezmann has scored 31 times for Atleti this term in all competitions, helping to keep his side's season alive on two fronts as they go in search of success in La Liga and the Champions League.
United scouts are closely monitoring the Frenchman, according to The Sun, and will make an approach this summer as part of a summer revamp at Old Trafford.
It is claimed that rumoured incoming Red Devils boss Jose Mourinho is a huge admirer of Griezmann, who the English club have placed alongside Borussia Dortmund's Pierre-Emerick Aubameyang at the top of their transfer wishlist.
Griezmann still has four years left to run on his deal at the Vicente Calderon, and Atletico are unlikely to let him go for cheap after he scored another crucial goal in Saturday's win over Rayo Vallecano.