‘The Special One’ worked with the Germany international while at the Bernabeu, and is known to be a huge fan of his abilities and character.
Khedira is currently returning to action following a long-term knee injury and looks set to struggle for a starting berth when he regains full fitness due to the presence of star names such as Luka Modric.
This has prompted talk that the 27-year-old could be allowed to leave, which has alerted Mourinho.
The Portuguese coach has often complained about a lack of depth and experience in his squad, and sees Khedira as an ideal partner in the centre of midfield for Nemanja Matic.
Real are however not willing to let their man leave cheaply, with a valuation of around £35m having been slapped on his head.
Despite a big money offer for Diego Costa of Atletico Madrid being mooted, Chelsea could potentially afford to fund a move for Khedira, with David Luiz being linked with a switch to PSG or Barcelona.