The White Hart Lane club even reportedly sent scouts to watch his performances and are believed to be weighing up a summer approach.
However, The Sun claim Chelsea are confident that Hazard's friendship with compatriot Tielemans could be a deciding factor when battling it out with Spurs.
As for Hazard, Real are expected to be placed under a transfer embargo at the end of the summer after FIFA found the club had breached rules in regard to signing players under the age of 18.
Eden Hazard - Spotlight on
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Their two-window transfer ban is currently being appealed and will likely mean they are free to bolster their squad before the start of next season.
That could lead to Hazard being targeted by Bernabeu officials.
However, Real's appeal is expected to be eventually thrown out, meaning they will likely not be able to spend throughout 2017.