Mourinho, 53, was appointed United’s new boss last month after Louis van Gaal was sacked.
Giggs had worked for two years as assistant under Van Gaal, with the original plan to promote him to manager once the Dutchman’s contract had ended in 2017.
But Van Gaal’s failure to secure Champions League football for next season forced the United board to act.
According to The Times, Mourinho believes Giggs distanced himself from both Van Gaal and his predecessor David Moyes at critical stages of their management.
The report claims Mourinho wants a loyal coaching staff behind him and doesn’t think Giggs can provide that.
Giggs was said to have been offered the role as Under 21s coach at United – widely seen as a demotion after two years as second in command.
Rui Faria, Mourinho’s assistant at Chelsea before the boss got the sack last December, is tipped to join the former Real Madrid and Inter Milan head coach at Old Trafford.
Mourinho has already been hard at work restructuring United’s team ahead of next season, with Villarreal’s Eric Bailly bought for £30million.
Sweden’s Zlatan Ibrahimovic should also join up after Euro 2016 on a free transfer, while Mourinho is seeking to complete a number of other deals