Shaw has struggled for regular first-team football at Old Trafford and has played second fiddle to Ashley Young for much of the 2017-18 campaign.
A disappointing season was compounded on Sunday when he was taken off at half-time of the win over Brighton, with Mourinho revealing his opinion that Shaw does not have the required personality to succeed as a Manchester United player.
And Le Tissier, who remains an iconic figure at Shaw's former club Southampton, is perplexed at Mourinho's handling of the England defender, and is expecting the player to depart the club in the summer.
He told Sky Sports: "I think if Mourinho is still there then it is inevitable Shaw will leave this summer. I have no idea what he is trying to achieve.
"There is just a really strange relationship going on there and I don't understand what he's doing because he's just destroying the player at the moment.
"I'm not sure if it was anything to do with him choosing Manchester United over Chelsea despite being a Chelsea fan when Mourinho was manager at the time.
"You also don't know what happens at training or how he lives his life to give an accurate assessment but, while there are players that this sort of approach can work with, I just don't think Shaw is one of them."
Le Tissier also tipped Shaw to join Mauricio Pochettino's Tottenham, with Danny Rose potentially heading in the opposite direction.
He added: "Having worked with Pochettino before and spoken highly of him, it would seem obvious for him to go to Tottenham with Danny Rose potentially going the opposite way as he is obviously desperate to get back up north.
"I don't think Southampton could compete in the wages department and the player would probably look at it as a backward move, so he would probably think he is worthy of playing for a club competing for the Premier League title."
And the Saints legend believes that the 22-year-old has struggled to recover from the broken leg he suffered against PSV back in 2015, but feels Mourinho should be doing more to help get the player back to somewhere near his best.
"I remember him getting subbed quite a bit at Southampton and there was a perception that he didn't take a knock very well and was a little bit on the soft side," he said.
"The injury is a big thing and he does seem a bit mentally fragile but, be it due to the injury or not, as a manager you've got to find a way to get the best out of him and I don't think Mourinho has done that.
"It is such a shame because he was such a talent at such a young age and the injury has set him back. As an attacking full-back, the boy had so much going for him.
"He's shown enough potential to be England's first choice left-back but it has been a long time since he's reached those heights, so he's got a lot of work to do in terms of getting games and being talked about at international level again.
"He's not going to do that at United so the only way is finding a new club. I'd be very surprised to see him getting in the England World Cup squad and it says it all that Mourinho sees a right-winger turned left-back in Young as more valuable."