Bale has struggled to win over Real fans in recent years due to his injury woes.
And the Wales international hasn’t enjoyed the best of starts to the new season with just two goals in the opening five La Liga matches.
Since moving to the Bernabeu from Tottenham in 2013 Bale has missed one in every three Real matches through injury.
Marco Asensio’s meteoric rise has also challenged Bale’s place in the starting line-up.
And the Spanish giants are said to be preparing for life without the 28-year-old next summer.
Spanish outlet Don Balon claim Real president Florentino Perez has already drawn up a list of players he wants to bring in next summer.
And Zinedine Zidane has made a personal plea to the club to bring Hazard to the Spanish capital.
It is claimed Zidane believes Hazard is the ideal replacement for Bale should he leave next year.
Real were heavily linked with a move for the Belgium international during the summer but an official offer was never made.
However, Hazard fuelled speculation about a possible move to Real in the future when he admitted in an interview last month that he would be open to the idea of playing in La Liga.
“For the moment I’m good in England,” Hazard told Marca.
“I play for one of the best clubs in England, but the Spanish league is special. Why not? For the moment I’m here.”
And on Real Madrid, he added: “I think they (Real Madrid) are now the best team in the world because they have won everything. They have good players, a good manager.