The Spanish giants have made Hazard their No1 target for the summer.
Hazard is stalling on signing a lucrative new contract at Stamford Bridge, which is being interpreted as a clear sign that Hazard wants a move to Real Madrid.
The Spanish giants are set to overhaul their squad after falling 20 points behind Barcelona this season.
They are also set to part company with head coach Zinedine Zidane, who Hazard is known to be keen to work with.
Manchester United have wanted Bale for the last two seasons but his former club Tottenham have first option – even though a deal is likely to be out of reach for all but the richest clubs.
United now seem to be out of the running for Bale after signing Alexis Sanchez from Arsenal last month leaving the way almost clear for Chelsea to lure him back to London.
Hazard and Thibaut Courtois have been in protracted contracted talks with Chelsea to extend their contracts beyond their current 2019 and 2020 expirations with Real reportedly interested in both players.
In November, Hazard told French television channel Canal+: “Everyone knows the admiration I have for Real, but for now I’m a Blue.”
Hazard’s father Thierry told Belgian paper Le Soir last year: “What I can say about Eden is that he refused an extension of contract to be able, if necessary, to follow the interest of the Real, where he would see himself well.
“But, as I speak to you, there is no contact. Eden is only one of the parties to the contract.”
Bale has had a difficult season in the Spanish capital with three separate injuries keeping him out for nine weeks over the course of the campaign.
The Wales international has nine goals in 21 appearances this season but has struggled to win the adoration of the Bernabeu faithful since his £85 million transfer in 2013.
Bale continues to live in the shadow of the king Galactico Cristiano Ronaldo who scored his 100th and 101st Champions League goals in the 3-1 Champions League last-16 first leg win over PSG on Wednesday.
The ex-Tottenham winger has won 12 trophies since joining Real including three Champions Leagues and La Liga.
