The Red Devils are understood to be keen on a £90m move for the 25-year-old next summer, as long as they return to Champions League football.
Real have targeted Blues forward Hazard as the perfect replacement, should Bale want a return to the Premier League
The European champions are long-term admirers of the Belgium international, losing out on signing him from Lille as he favored £35m move to Stamford Bridge.
The 23-year-old has since been a key member for the Blues, scoring 35 goals and assisting 40 in his 126 outings.
If reports in Spain are to be believed, Madrid will have to break the world transfer record again to tempt Chelsea to sell, having previously smashed it to sign Bale (£86m) from Tottenham in 2013. /End/