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 source dismissed reports by Spanish daily Marca which said the Welsh 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.
The 77-time capped Welsh 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.
Real boss Zidane said last weekend he thought it would be "best for everyone" if Bale's departure could be arranged quickly -- comments Bale's agent branded "disrespectful" of a player who helped the side to a Spanish league title and four Champions Leagues since arriving from Tottenham six years ago.
Zidane then insisted on Monday he meant no disrespect before Bale came on as a substitute in a midweek 3-2 friendly win over Arsenal in the United States.
"He had a good game and I'm happy for him," Zidane said.
"I do not know what's going to happen, for now he's with us. It did not change anything," he added.
Bale had been told by Real coach Zinedine Zidane he does not form part of his future plans at the Bernabeu