To show his real desire for leaving (and saving Levy at least 10 mil in loyalty money). Transfer request is a difference between "Meh, if i can i will leave but if i stay, no biggie " and "I want to leave no matter what". Remember how Modric did it? Transfer request, went on strike and Levy sold him for 30mil.
Also, as Pexbo perfectly said; Levy is famous as being tough negotiator. Which is all good. But he is in danger in crossing a line here. If he refuses to sell a club legend who gave more than 10 years to that club, then all players will think twice before signing for Spurs or signing long term contracts. Especially players with high potential whos goal is to play for giant clubs later.
Why sign for club who will kill your possible dream move later in career?