Left Back seems the obvious one, not necessarily because Jose might believe that Shaw is a lost cause but mainly because, in my opinion, the backbone of every top side is its stable defense. The goalkeeper, the back four and the defensive/holding midfielder should be sorted out by the manager and they should be the constant which will provide us the invaluable safety we need in order to be more flexible, dominant and adventurous in the middle and the attacking third. Right now, we don't only have a problem in the LB position, we have a huge hole there which creates problems in our game. Any opposition manager can put a pacey winger in that area and suddenly the whole team drops off several yards just to defend properly (the CB keeps an eye to cover the LB because he doesn't trust him, the defensive midfielder drops in the box to cover for the CB and this results to the whole team losing ground on the pitch).
What do we need? A very solid left back defensively, anything else would be an extra. Someone able to defend well in 1v1 against all kinds of wide players: pacey wingers who like to hug the touch line, explosive inverted wingers/forwards whose acceleration and creativity will test the FBs ability to defend on his weaker foot and, of course, wide play makers who search for pockets of space in those areas to make things happen.
If you take Real Madrid and their abundance in talent out of the equation, most Mourinho sides are built with one of the sides, preferably the left one, assigned with the task of doing all the heavy lifting in terms of creativity in the attacking third. It's quite rigid too with the forward dropping in between the lines to play with his back to goal, the winger staying wide and the attacking midfielder moving slightly to the left. Mourinho wants from his FB to stay a bit behind and create a diamond shape with the other three in order to have more passing options. So, overlaps/underlaps and drilled crosses come as a plus in our current predicament. Don't forget that Mourinho has won things with a right footed player at LB (Azpi) or with a left footed CB (Chivu) doing the job. We've seen this being implemented with Martial/Rashford staying wide to receive the ball, Zlatan dropping deeper and Pogba moving to the left with the FB behind them.
Central Midfielder, the player in front of the back four should also be a priority. When it comes to this player i agree with the notion that we need someone who possesses the required positional awareness and the ability to read the game well in order to keep both Pogba and Herrera in check. Both Pogba and Ander become more valuable to us when they spend more time in the middle and the attacking half but for different reasons. The former has the vision and the technical skills to create while the latter's ability to press and regain possession high on the pitch can potentially become a big ace up our sleeves in modern football. They could both benefit from having another player behind them.
Where i don't agree with the majority of the Caf is that we need a defensive "monster" to play there. I think that Pogba and Herrera can provide us with the engine to win the midfield battle. What the have failed to provide thus far is creativity from deeper areas, someone who will get on the ball and play the right pass that will allow us to move the ball smoothly and quickly out of our third. Mourinho has claimed that his ambition is to build a dominant side at OT but the sheer reality of things is that most of our attacking players look much more threatening when the have spaces which they can exploit while running with the ball and not when they have to create these spaces with their off the ball movement all the time. So, my pick would be a deep lying play-maker for that role instead of a pure defensive midfielder in the mold of Makelele.
Center Half. One that can defend well in the air so when we block out all the routes down the central channels, we can be sure that we will deal with the crosses coming into our box. Also one with a nearly perfect injury record. I am not greedy and i will not ask for ball playing abilities but i will ask for someone whose clearances/interceptions will be towards areas where our players are most likely to pick up the ball. This has always been the strongest suit of the Italian CBs who have mastered the art of defending (very) deep in their half. And before you say this should not be a priority please consider that, according to Squawka, we average nearly ten more defensive actions per game than Chelsea who also defend deep and play the counter under Conte. This indicates that the ball stays at their feet of our opponents in our third of the pitch more than it probably should. It partially explains why we have problems with moving the ball quickly out of our defense too.
I can fully understand those who think that we should acquire a game changer whose dribbling skills and decision making will immediately improve the abilities of our strong left side. But Martial has more than enough talent and the club has invested a lot in his abilities. Plus Zlatan looks like he will be here next season too, so both Martial and Rashford will get playing time on the left more than anywhere else on the pitch.
Having said that, i don't think Mourinho plans on buying a new central midfielder and have him in the starting lineup. The rumors about Griezmann come on a weekly basis nowadays and we can't simply ignore them. It makes sense for Mourinho to look for a player like him who will constantly look for runs on Zlatan's shoulder whenever the latter drops deeper or look to exploit the spaces Mourinho's system creates on the right side for his attacking players.
This will put immense pressure on Herrera-Pogba to deliver the goods in the midfield. Imho it's a huge risk in the most important area of the pitch since there are no guarantees that both of them will improve their overall game (and we need both of them to to do that) but Jose sees them in training every day so i will trust his judgement, if this happens.
But i think that he will go for a LB, a CB and Griezmann as options for the staring lineup and he will play his favorite 4231. In this scenario an addition like Dier makes sense too. An extra option in the midfield who can also help the manager revert the system to a 433 should the circumstances require such a thing.