The Wales international winger is reportedly close to sealing a world-record £85m switch to the Spanish side, after being linked with an exit from White Hart Lane for most of the summer.
The Mirror quotes Sheringham as saying: "You cannot blame the player for wanting to go to Real Madrid. Zinedine Zidane made the point last week – it is an opportunity that comes maybe once in a lifetime.
"And they are probably the biggest club in the world. It would be very difficult to turn down the chance to play there. And he is still improving. He's only 24 and has his best years in front of him.
"No-one would begrudge him the chance to go there. And I think most Tottenham fans would not deny Bale this opportunity. It is also an incredible amount of money to receive for one player."
Bale scored 21 goals in the Premier League last season.