Sources close to PSG have indicated the Blues’ co-owner Todd Boehly is trying to put together a deal for the 24-year-old, who seems certain to leave Paris either for a fee this summer or for nothing next year.
Spanish champions Barcelona have also been actively pursuing a player swap deal, it is understood.
The French club are also anticipating a renewed attempt from Saudi club Al Hilal to persuade Mbappe to agree personal terms.
He reportedly refused to even meet with an Al Hilal delegation in Paris last week. The Saudis were given permission to speak to Mbappe by PSG after lodging a world record £259million bid earlier this month.
PSG fear all efforts to sign Mbappe this summer will be in vain, though.
Sources close to the club are convinced he has already agreed a deal to sign for Real Madrid on a free transfer and secure a 160m euro (£138m) signing-on fee.
The Saudi league is still understood to be pushing hard to secure Mbappe as its marquee player, having appointed former Chelsea technical director Michael Emenalo as the director of football at its Player Acquisition Centre of Excellence (PACE) to lead recruitment centrally for the league’s clubs.
Speculation around Mbappe increased in intensity in June when it became clear he would not activate the one-year extension in his PSG deal to keep him under contract until 2025.
His exit from the club seemed almost certain when he was omitted from the pre-season tour of Japan and South Korea, which sparked the rush of interest in him, including the world-record bid from Al Hilal.
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