Media outlet Don Balon say the Blues have proposed an audacious swoop for the Wales international.
The Blues had a shocking start to the new season and slumped to a 3-2 home defeat against Burnley on Saturday.
Having offloaded an array of fringe players, Antonio Conte now desperately needs to strengthen his squad before the transfer window shuts.
It’s suggested the Italian chief still has around £120m to spend on fresh faces.
Don Balon say Manchester United are still sniffing around – and Bale fancies a move to Old Trafford – but Chelsea are ready to seize the initiative.
Real are keen on Stamford Bridge star Eden Hazard.
“Real Madrid star Gareth Bale is wanted by Chelsea ”
And Don Balon say the Premier League champions would be willing to use the Belgian as a sweetener to sign Bale.
Journalist Ian McGarry told The Transfer Window podcast that a Bale and Hazard swap could be on.
He said: “I think Bale has been subtly marketed by Jorge Mendes, the agent, who does most of Real Madrid’s buying and selling, to potential suitors in case they got a bite.
“The bite that would satisfy Real Madrid would be to sell him for the same price they paid, around £89m.
“The sticking point in this is Gareth Bale, he is a player who loves it at Real Madrid, does not see himself playing at another club this season.
“The base line is Madrid don’t see Bale as indispensable, Bale sees himself staying at Madrid, so you’ve got the rock and the hard place there.
“You bring in Hazard who scores 10, 12, 14 goals a season, leading assists in the Premier League, generally 10, 12, 14 a season as well.
“That’s serious competition for Bale who has had his injury problems, who hasn’t been as consistent as some expected him to be so I think there are questions to be asked.
“If Hazard tells Chelsea ‘I’m desperate to go to Madrid’, would they think about Bale as a replacement?”
Bale has been in Madrid since a 2013 switch from Chelsea’s London rivals Tottenham.