Bale, 30, who has been in the Spanish capital since 2013, was expected to join the Chinese Super League outfit on a three-year-deal worth £1 million ($1.1 million) a week.
The 77-time international had been told by Real coach Zinedine Zidane he does not form part of his future plans at the Bernabeu.
The former Tottenham Hotspur winger came off the bench to play half an hour in Real’s record 7-3 defeat to rivals Atletico Madrid on Saturday.
The source dismissed reports by Spanish daily Marca which said the player’s family had blocked a potential move to Asia.
Marca said those close to Bale had asked his representatives to look for clubs in Europe “before definitively saying yes to Jiangsu Suning’s offer”.
The Chinese transfer window closes on Wednesday.