Hazard has only scored five times since leaving Chelsea two years ago in a deal worth up to £ 130million.
Injury issues mean the Belgian has only played 43 games in all competitions in two seasons in Spain.
The 30-year-old was not in the squad for Saturday’s game against Villarreal as Real finished the campaign second in La Liga, two points behind champions Atletico Madrid.
Real Madrid, who lost to Chelsea in the Champions League semi-finals, ended this season without a major trophy for the first time since 2009-10.
The collapse of the European Super League proposal, along with their continued interest in Mbappe and Haaland, means Real Madrid need to raise funds through player sales this summer.
In addition to Hazard and Bale, Real will listen to offers from other players such as Luka Jovic.
Hazard has won six major trophies, including two Premier League titles, in seven seasons at Chelsea.
Meanwhile, Bale spent the 2020-21 season on loan at Tottenham, scoring 16 goals and ending the campaign with a brace at Leicester on Sunday.
After that match, Bale said he had made a decision on his future but would wait until the end of this summer’s Euros before going public as it “will cause chaos”.
The Wales captain’s contract with Real expires in the summer of 2022.