The timing of Mourinho’s reported contact with United is interesting as frustrated Old Trafford fans booed their team off the pitch in the aftermath of their latest setback.
According to the report, the Portuguese coach used the letter to pledge his commitment to adapting his style of management and how he would overhaul the squad.
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Mourinho has been linked with the United job before when Alex Ferguson left to be replaced by David Moyes in 2013.
As well as two spells at Chelsea, Mourinho has also coached Porto, Inter Milan and Real Madrid.