But the Daily Mail claim Real have been encouraged by Hazard to bid and will make a £150m offer in the coming weeks.
And the European champions believe they can lure the Belgium captain, who has two years left on his Chelsea contract, to Spain.
Mbappe has arguably been the stand out attacking player at the World Cup in Russian Federation, despite only being 19-years-old and rumours have been circulating for some months now that PSG may be forced to pull out of the deal they agreed with Monaco at the end of last season.
"I'm doing well in blue so my future does not bother me".
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Chelsea have been keen to tie him down to a new deal, but the Belgian has so far refused to sign a new contract, with the spectre of Real Madrid always looming in the background.
Their hopes of keeping their star man could be boosted soon.
Chelsea have not made any summer signings and their managerial situation is still uncertain with Antonio Conte in charge.
And that could see the Blues complete deals for Aleksandr Golovin, Daniele Rugani and Jorginho - the trio having all been heavily linked to join Sarri in the English capital.
The Red Devils face England in a third-placed play-off on Saturday (3pm) after losing to France in the semi-finals.