The Mirror says former Blues manager Mourinho blamed the Germany midfielder for getting himself sent off in Real's shock defeat to Levante earlier this month.
That was one of only two starts the 24-year-old has made since the opening of the La Liga season, although he did play in Tuesday’s Champions League win over Ajax.
With Mourinho seeking to introduce his pair of Turkish summer signings, Hamit Altintop and Nuri Sahin, and Brazilian Kaka showing glimpses of a possible return to his best form, Khedira could become surplus to Bernabeu requirements.
Khedira would not be eligible for the knockout stage of the Champions League if he did join Chelsea in January.
But that situation did not prevent Blues owner Roman Abramovich sanctioning the £25million signing of Brazilian defender David Luiz from Benfica in the last winter window.