The French youngster has been in impressive form this season, scoring 15 goals in 23 La Liga appearances this term - including one against Real Madrid in Real Sociedad's 5-1 defeat back in November.
He's been linked with a move to Arsenal in recent months, as well as Paris Saint-Germain and Juventus.
The paper reports that Griezmann would prefer a move to the Bernabeu over Arsenal or any of his other suitors, and that the Spanish giants' president Florentino Perez is keen on getting the deal done.
The news of Madrid's interest will come as a blow to Arsenal who are likely to be in the market for a new forward in the summer.
The Gunners can only currently call upon Olivier Giroud to lead the line, with Yaya Sanogo and Nicklas Bendtner providing support.
Real Madrid are in action in the Champions League against Schalke this evening, while Arsenal are 2-0 down in their last 16 clash against Bayern Munich from the first leg.