Bale returns to his former club on a one-year loan deal, with British media reporting that Spurs will pay 20 million pounds ($25.8 million) in wages and fees for the 31-year-old Wales forward.
Reguilon, who spent last season on loan at Sevilla, arrives for a reported fee of 30 million euros ($35.51 million).
Bale made more than 200 appearances for Spurs between 2007 and 2013, scoring 56 goals with 58 assists before his then world record move to Real for 100 million euros ($118.37 million).
He won four Champions League trophies and two La Liga titles with Real but has divided opinion amongst the club’s fanbase, with some supporters frustrated by a perceived lack of commitment from the Welshman and his poor injury record.
Bale was poised for a move to Chinese club Jiangsu Suning last year before the deal fell through.
He endured his worst season at Real in 2019-20, scoring only three goals in all competitions, and barely featured in their run to the Liga title.
His growing rift with manager Zinedine Zidane intensified when he did not travel with the squad for a Champions League clash against Manchester City last season.
The signings of Bale and Reguilon follow the arrival at Spurs of Denmark midfielder Pierre-Emile Hojbjerg, Ireland defender Matt Doherty and former England goalkeeper Joe Hart.