To do so, 'Don Balon' claim 'Los Blancos' will sell Gareth Bale in order to fund a move for the Atletico Madrid star.
They are willing to let Bale leave the Bernabeu for £133m and use the cash to trigger Griezmann's £90m release clause.
By signing the 26-year-old, Real would not only take away one of their fierce city rivals best players, but also frustrate Barcelona's transfer ambitions.
The Catalan side, as well as United and Paris Saint-Germain have all registered an interest in Griezmann, but it is Real who are leading the way in the race for his signature, given their recent Champions League success.
Manchester United are understood to be one of the teams monitoring Bale, but they reluctant to pay over £130m for such an injury-prone player.