The nine-times European champions have a reputation for getting their man and sports daily Marca were sourcing radio station Cadena COPE when they said a deal had been agreed and would be confirmed on Friday for 99 million euros (85 million pounds).
Real's signing of Cristiano Ronaldo from Manchester United in 2009 set the world record transfer fee at some 96 million euros.
Real play a friendly against Qatar side Al Sadd, the annual Santiago Bernabeu trophy, on Thursday evening when the club will be celebrating the return of their all-time leading scorer Raul. They are unlikely to want to trump that event with the Bale signing.
Even if the deal for the Welsh international is not confirmed on Friday, all the indications are that the player will be running out at the Bernabeu this season in the currently vacant number 11 shirt.
The 24-year-old has been ruled out of action for the opening matches of Tottenham's season as he has suffered a couple of niggling injuries, though media speculation has suggested it is part of a tactic to force through the move.
His current club have been busy in the transfer market, landing Spain striker Roberto Soldado from Valencia for a club record 30 million euros and are reportedly close to exceeding that figure with a move for Brazilian forward Willian from Anzhi Makhachkala.
The transfer window shuts in England on September 2.
(Reporting by Mark Elkington, editing by Patrick Johnston)
