The Spanish club are set to spend a world-record £105million on the Wales international.
And while White Hart Lane legend Sheringham has no issue with Tottenham selling Bale, he wants the cash used to strengthen the squad of manager Andre Villas-Boas.
"You cannot blame the player for wanting to go to Real Madrid," Sheringham told the Sunday Mirror.
"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.
"Other players have gone to Real from the Premier League - players like David Beckham, Steve McManaman and Michael Owen. But Bale is younger and still to reach his peak. 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."