Express Sport understands Griezmann is Manchester United's No1 transfer target for the summer.
With Zlatan Ibrahimovic and Wayne Rooney in limbo over their future, the Red Devils are keen to sign a marquee striker.
And the Manchester Evening News claim sources have told them United are close to completing a huge deal for the France international.
It is said that manager Jose Mourinho is ready to match Griezmann's release clause of £86million to land his man.
Manchester United are close to completing a deal for Antoine Griezmann
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
People close to the player believe a deal is all but done.
United want a new-look strikeforce next season as they aim to challenge on all fronts.
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Mourinho even considered signing Griezmann in January but ultimately opted against a move.
Instead, he is confident of landing the Atletico Madrid star at the end of the season.
Jose Mourinho appears to be close to landing his top target
Atleti boss Diego Simeone admitted earlier today it would be hard to keep his prized asset.
"Yes, because there is a termination clause. There is no security," he told Marca.
"From now on I do not know what can happen. He would never let me tell him to stay. I did not do it with [Diego] Costa or [Radamel] Falcao.
"To the players who have given me life I accompany them in the decisions that they make. But it is clear that for as long as he stays, the team and the club will continue to grow."
United are also interested in Everton's Romelu Lukaku but are yet to push ahead with a bid.