According to Diario Gol (via Calciomercato), Asensio would be open to a switch to the west London outfit, and Real could offer the 22-year-old in part-exchange for Hazard, with whom they have long been linked, in the summer.
Asensio made his breakthrough at Real last term and topped off an impressive season by scoring in the UEFA Champions League final.
He then netted in both the home and away leg as Real brushed Barcelona aside in August's Spanish Super Cup to prompt predictions of a wonder season for Asensio and the Madrid giants.
Neither have come to pass. Real have struggled in La Liga and are fighting to keep hold of a Champions League qualification spot as they lie in fourth, 19 points back from leaders Barca.
Meanwhile, Spain international Asensio has not scored in the Spanish top flight since November 5 and has not started a league game for Los Blancos since December 9.
Real manager Zinedine Zidane seems to have lost faith in the young attacker, who is capable of playing through the middle or on either flank.
He remains a relatively raw talent, though, and could be an ideal long-term replacement for Hazard, 27, at Chelsea.
Should the Belgium international push for a move to the Santiago Bernabeu, Chelsea would do well to try to have Asensio included in the deal.
However, despite his woes for much of this season, Real may be reluctant to let Asensio go, as he has long been viewed as a key part of the next generation of Los Blancos stars, per Spanish football writer Andy West:
He is far from the only Real player to have endured a relatively poor 2017-18 campaign, but he is young enough to be excused for a drop in form, and there should still be confidence he will return to his best again.