Cameras showed the Belgium international laughing with former teammate Kurt Zouma and goalkeeper Edouard Mendy on the pitch after Madrid's dream of winning the competition for the 14th time was brought to an end following Wednesday's 2-0 defeat.
Madrid fans took to social media to criticise Hazard, and ESPN has learned that the anger has extended to several club members and even the changing room.
A club source told ESPN that it was "intolerable" that Hazard showed "that attitude after an elimination" so painful for Madrid.
The same source reproached the forward for "lacking tact" following such a poor performance against his former club.
Hazard, 30, started in the semifinal return leg to lead Madrid's attack alongside Karim Benzema but was anonymous throughout the game and replaced by Mariano in the 88th minute with his team losing 2-0.
"Eden has enough experience to not have that attitude when the cameras are on him," a source told ESPN before giving the example of another former Chelsea player, international teammate Thibaut Courtois, whose display in front of the cameras was more somber.
Hazard arrived at Madrid in the summer of 2019 as a record signing after seven successful seasons at Chelsea and having starred for Belgium at the 2018 World Cup. Madrid coach Zinedine Zidane saw Hazard as a cornerstone for his new project.
However, Hazard has played only 37 games in two seasons at Madrid and failed to be consistent after picking up 11 injuries since his arrival.
Spanish media predict that his laughs after the team's elimination by Chelsea could be the end of his career at Madrid. However, the club has closed ranks after a very difficult night in London.
The club source insists that "it wasn't the time or the place to give that image" but believes that everything will be settled by giving Hazard a "slap on the wrist."
Despite the defeat, the club puts a lot of value on the fact that the team reached the Champions League semifinal after making no signings last summer.
Although Chelsea were superior over the two games, having held Madrid to a 1-1 draw in the first leg, the Spanish club still had a chance to take Wednesday's game to extra-time until Chelsea scored their second five minutes from time.