Express Sport understands Griezmann is Jose Mourinho's chief target for the summer.
Manchester United are eying summer reinforcements after a disappointing season that sees them currently sixth in the Premier League table.
Their best hopes of securing Champions League football lie in the Europa League, with Mourinho hinting he could prioritise Europe over their domestic campaign.
Antoine Griezmann has told Manchester United what they must do to sign him
Antoine Griezmann's next club
Wed, April 5, 2017
Which club is favourite to sign Atletico Madrid ace Antoine Griezmann?
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Which clubs are tipped to sign Atletico Madrid ace Antoine Griezmann?
And the Sun claim their hopes of luring Griezmann hinge on how the rest of the season plays out.
The Atletico Madrid striker is said to have told the Red Devils he will only join should United book their place in Europe's top club competition next season.
Paul Pogba's influence has proved key to the deal but the midfielder's own problems could deter Griezmann from making the move.
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Manchester United boss Jose Mourinho is determined to sign Antoine Griezmann
However, United's loss could be Chelsea's gain.
Antonio Conte is determined to sign the forward amid fears over Diego Costa's future.
And, with Champions League football all but sealed, the Blues could make Griezmann their marquee signing of the summer.