The Belgium international has been linked with a potential switch to los Blancos ahead of the 2018 summer transfer window.
Hazard has endured a frustrating season at the west London club after Chelsea FC were unable to defend their Premier League crown.
The 27-year-old has scored 11 times and has made four assists in 28 games in the Premier League this season.
Chelsea FC are in fifth place in the Premier League table and 10 points behind Liverpool FC and Tottenham Hotspur.
Hazard has been regularly linked with a move to Real Madrid in recent seasons and manager Zinedine Zidane is the Belgian’s football idol.
Wales international Bale is also facing an uncertain future at The Bernabeu amid reports he could face the axe at Real Madrid this summer.
Former Liverpool FC star Redknapp believes Chelsea FC should look to sign Bale if Hazard does leave the defending Premier League champions at the end of the season.
“Should he leave for Real this summer, they would need a new A-lister, and negotiating a swap for Gareth Bale would be a great solution for Chelsea,” Redknapp told the Daily Mail.
“Every summer there seems to be talk of a big-money return to England for the Welshman and Bale, now 28, would be a perfect fit for Chelsea.
“He is a big name and someone who already knows the Premier League.
“Jose Mourinho’s Manchester United would no doubt be in the mix too but if Chelsea can play a clever game with Real and Hazard this summer, then they can make it work in their favour.”
Hazard has won two Premier League titles, the Europa League and the League Cup since his move to Chelsea FC from Lille in the 2012 summer transfer window.
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