The German international recently rejected the offer of a new contract at the Bernebeu in favour of a summer move away from the Spanish capital, yet a deal could not be worked out.
Real are convinced that their man has not yet got an agreement with any other side, and are still hoping to convince him to stay.
But Arsenal and Chelsea are both interested in the World Cup winner.
The Gunners were closely linked with Khedira during the last transfer window, yet claims that a transfer was close were deemed to have been wide of the mark by Arsene Wenger.
The rangy enforcer could be tempted by life at the Emirates Stadium, where he would join up with compatriots such as Lukas Podolski, Per Mertesacker and Mesut Ozil.
Likewise he could link up with a figure from his past at Chelsea.
Jose Mourinho coached Khedira while in charge of Real, and is known to be a massive fan of the ex-Stuttgart ace.
The player himself is after a massive contract, but the lack of a transfer fee may tempt the two English sides.
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