The Spanish paper claims that the deal has finally been agreed with Real paying €84m (£71.7m) with a further €7m in add-ons taking the final price to €91m (£77.6m)
Real Madrid paid Manchester United €96m for Cristiano Ronaldo back in 2009 – at the time that converted to £80m.
Throughout the lengthy transfer negotiations between the two clubs it had been reported that Tottenham were looking for a transfer fee of around £100m which would have smashed the world transfer record.
The paper also published a photo on its website of what it said was a box Adidas boots for Bale sent by courier to Real's Valdebebas training facility.
Spurs have already been busy spending the expected windfall from Bale's transfer and on Friday they broke their transfer record for the third time this summer with the signing of Roma's Erik Lamela.
The Argentina international has joined the club for a reported fee of £30 million – surpassing the £26m paid for Valencia striker Roberto Soldado and, before that, the £17m spent on Brazil midfielder Paulinho.
Lamela was Tottenham's second signing of the day having followed Romanian defender Vlad Chiriches - who cost £8.5m from Steaua Bucharest - to the north London club.
The deal for Lamela takes Tottenham's summer spending past the £100m barrier.