On the evidence of the first three games the answer is very simple; No.
United have played Bournemouth away, Southampton at home and Hull City away and have conceded only one goal. More to the point, David De Gea has had very little to do.
This is in stark contrast to the last two seasons where the defensive statistics have been very good but only because De Gea was very good. Good enough, in fact, to win the player of the season award on both occasions. Without him the defensive statistics would have been quite poor.
The most usual pairing of centre backs last season was Chris “Bambi on ice” Smalling and Daley “defensive midfielder and not very tall” Blind. Marouane Fellaini was brought in to help out in a few games when the situation became a little precarious.
With different full backs tried almost every week it is hardly surprising that De Gea was one of the most overworked goalkeepers in the Premier League. Cameron Borthwick-Jackson, Timothy Fosu-Mensah, Matteo Darmian, Antonio Valencia and Ashley Young all played in the two wide positions with varying degrees of success.
Young was useless, Valencia was OK going forward but completely caught out when trying to defend, Borthwick-Jackson and Fosu-Mensah did little wrong but couldn’t be expected to maintain a high enough level of consistency throughout the season and Matteo Darmian was, basically, written off after a few games.
The centre backs last season, with a lot of help from De Gea, did OK. That is not good enough if you want to win the title.
The first signing made by Jose Mourinho was Eric Bailly. A tall, commanding centre half who can also play at left back. He has replaced the dithering Smalling alongside Daley Blind and the defence has looked transformed. Luke Shaw has returned from injury and slotted straight back in to both the United and England teams.
The question is; what happens if one or the other centre half gets injured? Who is just sitting waiting for his chance?
Number one on the list is probably Chris Smalling but, as he proved last year, he is hardly reliable. Constantly getting himself entangled, in the penalty area, with the opposing forwards means that, with the new rule, he would give away a lot more penalties this season than he did last. He is also quite adept at getting sent off, giving the ball away, being caught out of position and mis-timing tackles. Other than these few minor faults he is fine.
Marcos Rojo would likely be the next in line. He still hasn’t really fulfilled his potential since signing for United and has had good and bad games at both centre half and left back. If he can sort out his inconsistency he would be a good squad player to have around.
Phil Jones, when free of injury, plays like a headless chicken. Doing things like running around aimlessly and diving into tackles when he shouldn’t are traits that he should have lost by now. He needs to go out on loan or be sold. Either way he is no longer good enough for United.
I have to assume that Axel Tuanzebe is deemed too young at present to be considered but he is reckoned to be a great prospect for the future.
So now the answer to the question changes. Yes, they do need to buy another centre half. Having broken down the alternatives it is understandable that Jose Mourinho would like to bring in Jose Fonte from Southampton or Raphael Varane from Real Madrid. Even Fabinho from Monaco, who would be an ideal replacement for Antonio Valencia who, as previously stated cannot defend, would be a sensible purchase.
United currently have no problems with their defence, as their record shows. Injuries, however, to one or more key players would change the landscape. On this basis alone Jose Mourinho needs to bring in at least one new centre half if nothing else.
On a slightly different subject, here’s an interesting take from the Guardian on how tactics in the Premier League have evolved since the early 90’s.