The Red Devils, who finished seventh last term, have spent a reported £60m already this summer to secure the services of left-back Luke Shaw from Southampton and Athletic Bilbao midfielder Ander Herrera.
However, while BT Sport pundit Savage currently expects Louis van Gaal's side to improve upon last season's showing by challenging for a Champions League place, he believes that a defender, a midfielder and an attacker are needed if United are to stand any chance of winning a 21st league title.
"If you go on pre-season, you have seen a side playing with confidence and moving the ball quickly. They have a new shape and a new dynamic. They are two or three players short of making a realistic title challenge, though," said the former United trainee. "They still have some fantastic players there, but right now it is fight for fourth between them and Liverpool.
"They need a commanding centre-back and another midfielder and if you look at the forward areas, they've got Wayne Rooney and Robin van Persie, who are top-class footballers, but what if one of them gets injured? They have Javier Hernandez or Danny Welbeck to come in.
"Yes, Hernandez scores when he comes off the bench, but there are question marks over him when he starts, whereas does Welbeck score enough goals? Maybe not. They are three players away from being a top, top team. If they all of a sudden brought Angel di Maria, Mats Hummels and Arturo Vidal, we are talking about a different Manchester United and one that can challenge for the title.
"They've got Ander Herrera midfield and Darren Fletcher has had a great pre-season. It's a big season for Tom Cleverley. Can he prove his critics wrong? It would be nice to see him do that. Even so, they need a world class player in that area and Vidal would fit the bill."
United get their campaign underway tomorrow afternoon with a home encounter against Swansea City.
Manchester United vs. Swansea City is one of 38 Barclays Premier League matches exclusively live on BT Sport this season along with top games from the FA Cup, UEFA Europa League and live action from more top flight football leagues than any other UK TV provider.