Griezmann is understood to be Jose Mourinho's No 1 transfer target for the summer as the Manchester United boss aims to strengthen his squad.
It was claimed last month that the 25-year-old had already agreed personal terms with United, which would see him paid as much as compatriot Paul Pogba.
How Man United could look with Antoine Griezmann
Tue, February 28, 2017
Is this how Manchester United's squad could look with Antoine Griezmann
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Is this how Manchester United could look with Antoine Griezmann
But according to Spanish newspaper AS, Griezmann is 'extremely settled' at Atletico.
And the France international is said to be giving positive messages about his future behind closed doors.
Antoine Griezmann wants to stay at Atletico
As a result of Griezmann's inclination to stay at Atletico, United have set their sight on other targets including Kylian Mbappe and Robert Lewandowski.
Atletico manager Diego Simeone's decision to stay at the club next season despite interest from Inter Milan has also added to Griezmann's intention to remain at the Vicente Calerdon.
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Antoine Griezmann celebrates scoring a goal
Last month, Griezmann revealed he has 'many doubts' about a move to England saying: "England, there are many doubts about my private life. There is rain, bad weather… I need to be happy outside."
The report says that Griezmann is keen for the club to make signings in the summer.
Manchester United want to sign Antoine Griezmann
But they are currently banned from buying players by FIFA are breaching rules over the transfer of foreign players under 18.
Atletico have appealed to the Court of Arbitration for Sport along with rivals Real who were also banned.