Bale, currently on a straight loan this season with Spurs, has declared this week his intention to return to his parent club this summer.
However, El Confidencial says Florentino and his fellow directors are desperate for Spurs to keep hold of Bale as there is no place for the Wales captain in their plans for next season.
Instead, Florentino hopes to use his good relationship with Spurs chairman Daniel Levy to convince him to work on Bale about staying in London.
Bale has just over a year to run on his Real deal and Florentino would prefer he see out those 12 months away from the club.
"The main reason I came to Spurs this year was primarily to play football, obviously, but with the Euros coming up, I wanted to enter this tournament in great shape," Bale said earlier this week.
"The original plan was to only have one season with Spurs, then after the Euros I will have another year in Madrid. My plan is to go back. That's all I planned, to be honest."