The 31-year-old arrived at the Bernabeu with high expectations after completing a £103m move from Chelsea in 2019, but he has endured an injury-hit three years in the Spanish capital.
Hazard has only scored six goals in 66 appearances for Real Madrid in all competitions - as well as providing 10 assists - and he has missed a further 72 games through fitness or illness woes.
The 2021-22 season saw Hazard make just 23 appearances - scoring one goal and setting up two more in the process - and he started just seven La Liga matches in total.
A fibula fissure plagued Hazard's 2021-22 campaign, but he recovered in time to make the bench for the 1-0 win over Liverpool in the Champions League final.
The Belgium international has dropped behind the likes of Rodrygo, Vinicius Junior and Marco Asensio in the pecking order, while Federico Valverde is often preferred out wide too.
As a result, Calciomercato claims that Ancelotti is open to parting ways with Hazard this summer and has informed the Blancos board of his stance.
The La Liga and Champions League winners are now planning to put the feelers out to gauge interest in Hazard, who recently insisted that he wants to stay and prove his worth to the capital club.
"The coach knows that I still have two years here. The first three years have not gone well. I want to show everyone what I can do," Hazard told Het Laatste Nieuws before the 2022 Champions League final.
"I'm still in my dream, I've always dreamed of this. My life goal has been to play here. My family has always been happy here, [and] even now they are."
Hazard remains under contract at the Bernabeu until the summer of 2024, and a recent report claimed that Ancelotti planned to trial the ex-Chelsea man as a false nine next term.
Newcastle United were strongly linked with a move for Hazard before the transfer window opened, but it remains to be seen if the Magpies will reignite their interest.
Since arriving in Madrid, Hazard has won the Champions League, two La Liga titles and two Supercopa de Espana crowns with Los Blancos after winning six major trophies with Chelsea.
Real Madrid have already allowed Luka Jovic and Gareth Bale to depart the club this summer, while Asensio continues to be linked with a move to Newcastle, Arsenal, Liverpool or AC Milan.
Los Blancos begin their period of pre-season friendlies against Barcelona in the United States on July 23 before subsequent fixtures with America and Juventus later in the month.