Website Football Leaks published documents detailing the star's transfer in 2013 which state the world-record figure paid by Madrid to Tottenham.
It is claimed the Santiago Bernabeu club parted with €100,759,418 for Bale – more than the €94m fee paid for Cristiano Ronaldo in 2009 – despite their previous insistence that Bale's deal cost closer to €91.59m.
Reports in Spain also suggest Tottenham were forbidden from disclosing details of the transfer fee, which is understood to have been paid in four installments.
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Barnett is furious at the leak and has called for apologies from the Football Association, telling Telegraph Sport: "There should be an inquiry and an independent investigation because it's outrageous.
"There also needs to be an apology from the Football Association to the clubs and the player.
"I think it's disgraceful that people can get hold of this sort of stuff. It shows complete disregard for both clubs and the player."
A British journalist claimed this month that Madrid had instructed him not to mention Bale's world-record fee during an interview with the 26-year-old.
"Real Madrid are a club so worried about image that when I interviewed Bale it was requested to me, 'don't put in the article how much Gareth Bale cost'," Jonathan Norcroft of The Sunday Times told the BBC.
"The reason for that is because it was a world-record transfer fee, but Cristiano Ronaldo doesn't like to see that someone else cost more than him."