Manchester United defender has 'no excuse' for lacking commitment
He has only played 15 times for United this term and has not figured since the 1-1 draw
with Bournemouth on 4 March.
"It is all about discipline, dedication and desire," Mills told BBC Radio 5 live's Monday Night Club.
"It appears Luke Shaw isn't giving it his all. He has all the ability in the world but it all comes down to attitude.
"Even if he has been injured, there is no excuse not to keep the weight off or to be as fit as you possibly can be and look like you are making the biggest effort ever. It is your job."There is no excuse whatsoever."
Shaw signed for the club in June 2014 for £27mbut was criticised by then manager Louis van Gaal for lacking fitness.
He then broke his leg in two places during a Champions League game in September 2015, which kept him out of the game for almost a year.
In November, he was criticised by Mourinho for missing the Premier League win over Swansea after he said he was not able to play.
He was not part of the squad for Saturday's 0-0 draw against West Brom, despite United having a lengthy injury list.
"I think Mourinho has had enough and thinks he can't let this attitude poison any youngsters or drip feed into others," added Mills.
"Would Sir Alex Ferguson have stood for this? No. Fergie would have just got rid of him. Simple as that.
"Fans and managers will forgive you for missing a goal, for making a mistake or not having the greatest touch, as long as you work hard. That should be mandatory and a prerequisite for any footballer."
Former Republic of Ireland midfielder Andy Townsend told 5 live: "I don't think everybody can be wrong. This is a challenge for Luke Shaw now. He has to respond to this.
"If he wants to err on the side of public sympathy and act like he has been hard done by that is the wrong road.
"Mourinho is saying to him that he is well down the pecking order and if he wants a career at Manchester United then things have got to change."
BBC Sport's Simon Stone:
It is understood Mourinho has not spoken at any great length to the 21-year-old since Shaw was caught talking to a Bournemouth player by his manager during half-time of the recent ill-tempered clash at Old Trafford.
As Zlatan Ibrahimovic and Tyrone Mings had only just had the altercation that led to them both being banned, maybe there was a reason for Mourinho to be irritated.
But since then he made Shaw train with the United youngsters before his side's Europa League game against FC Rostov and did not pick him on the bench for Saturday's home draw with West Brom.
As Ashley Young started at left-back and Timothy Fosu-Mensah, Axel Tuanzebe and Matthew Willock - the latter two having made one substitute appearance between them - were among the substitutes, it does not say much for Shaw's position in Mourinho's plans.
The player is understood to be disappointment at his treatment, with some of his team-mates apparently feeling the same way, with those close to the former Southampton man privately questioning the wisdom of Mourinho going public with his criticism.
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