The Daily Star Sunday says the bidding for Atletico Madrid's French striker could reach £80m next year.
Griezmann, 25, parted company with his agents – Grupo Santos – last week and announced he will handle future deals himself, along with his father Alain and trusted aid Eric Olhats.
Manchester United already had him on their wanted list, but neighbours Manchester City would be keen, too, despite having top-gun Sergio Aguero and Brazilian sensation Gabriel Jesus arriving in the New Year.
City would be unlikely to leave the field clear for United if a top talent like Griezmann was available.
But the Old Trafford club face competition anyway for the six-goal leading scorer at this summer's Euros, where he was also named as the player of the tournament.