That’s according to journalist Eduardo Inda, who’s made the revelation while appearing on Spanish show El Chiringuito TV.
Griezmann will have plenty of options on the table as he gets set to leave Atletico Madrid at the end of the season.
And Barcelona are thought to be the frontrunners in the race to secure his signature once they’ve activated his £89m (€100m) release clause.
Reports last month suggested the Catalan giants already have an agreement in place for Griezmann to head to the Nou Camp.
Inda has now revealed Real are going to take make their own proposal though as they look to bolster their forward line.
“Madrid knows that there is a verbal agreement with Barcelona,” he told El Chringuito TV.
“They are going to try to make the product more expensive, which I think is very good.
“But Madrid is going to make an offer for Griezmann.”
Griezmann’s head may be turned if Real do swoop for him when the transfer window reopens.
The 26-year-old supported Real as a youngster and was a fan of former Bernabeu ace David Beckham.
Don Balon indicate Atletico would prefer to see Griezmann to join their city rivals than Barcelona too.