Bale, 30, had been expected to join Chinese Super League club Jiangsu Suning on a three-year deal, earning a reported £1 million a week, BBC Sport reported.
Last week Real boss Zinedine Zidane said Bale was "very close to leaving" having fallen out of favor.
Zidane added his exit would be "best for everyone."
Bale joined the Spanish club for £85 million from Tottenham in 2013 in a world record deal at the time.
He has three years left on his contract with the Bernabeu club where he has won four Champions Leagues, one La Liga title, a Copa del Rey, three UEFA Super Cups and three Club World Cups.
Bale scored three goals, plus a penalty in a shootout, in four Champions League finals for Real as the Spanish giant won the competition in 2014, 2016, 2017 and 2018.
However, injury problems have limited him to 79 La Liga starts in the past four seasons.
He played 42 matches for Real Madrid last season but was booed by the home supporters at times during the campaign.
Zidane's return as Real boss in March was described as "bad news" by the forward's agent, Jonathan Barnett, because the Frenchman did not want to work with Bale and the two men disagreed on playing style.
Bale ended last season on the bench as Real endured its poorest domestic campaign in 20 years, with 12 defeats, 68 points and a third-place finish – 19 points behind champion Barcelona. Real was also knocked out of the Champions League by Ajax in the last 16.
Bale has been reportedly nicknamed "The Golfer" by his Real teammates, and goalkeeper Thibaut Courtois said Bale did not attend a team meal because he did not want to miss his bed time.
He is Wales' record goalscorer with 31 goals in 77 caps.
Preseason to forget
Bale was left out of Real's first preseason game of Real’s US tour, a 3-1 defeat by Bayern Munich on July 21. After the game Zidane said, "We hope he leaves soon. It would be best for everyone. We are working on his transfer to a new team."
In response to Zidane's comments, Bale's agent Jonathan Barnett told AFP, "Zidane is a disgrace – he shows no respect for a player that has done so much for Real Madrid."
Bale came off the bench in Real's next game to score as they won on penalties after a 2-2 draw with Arsenal in the International Champions Cup, although Frenchman Zidane insisted "nothing had changed."
He again started on the bench as Real was thrashed 7-3 by Atletico Madrid in New Jersey, but came on for the final 30 minutes.