Express Sport understands Griezmann is Manchester Unitd boss Jose Mourinho’s No 1 transfer target this summer.
The Manchester Evening News claim United are on the verge of completing a deal for Griezmann.
But according to Transfer Market Web, Chelsea boss Antonio Conte is set to challenge Jose Mourinho for the France international.
Atletico Madrid star Antoine Griezmann is a wanted man
How Man United could look with Antoine Griezmann
Thu, March 23, 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
There is a termination clause. There is no security
Atletico Madrid boss Diego Simeone on Antoine Griezmann
It is claimed Griezmann is likely to leave Atletico Madrid after three impressive seasons and is also being tracked by Premier League rivals Arsenal and Manchester City.
Barcelona, Bayern Munich and Paris Saint-Germain are other top European clubs interested in the 26-year-old, although a move to Real Madrid has been ruled out given they are bitter rivals of Atletico.
Griezmann’s release clause at the Spanish club is reported to stand at about £95million, meaning his move is poised to break the world-record fee of £89m United paid to sign Paul Pogba from Juventus last summer.
Atletico boss Diego Simeone admitted his star man could leave this summer, saying: "Yes, because there is a termination clause. There is no security.
Antoine Griezmann is currently on international duty with France
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"That's what he told me last year, 'If you stay, I'll stay'. And I said, 'Yes, why would I not stay?' And they began to work with the representatives for everything that had to be signed. And he stayed.
"And 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.
"But it is clear that for as long as he stays, the team and the club will continue to grow."