The Red Devils agreed a £40m fee with the defending Premier League champions to secure the signature of the Serbia international.
Chelsea signed Tiemoue Bakayoko in a £40m transfer earlier this month but the AS Monaco midfielder started the 2017-18 with fitness problems.
Matic has already made an impressive start to his Manchester United career after being named man of the match against Real Madrid and West Ham United.
The Serbian midfielder helped the Red Devils to ease to a 4-0 victories over West Ham and Swansea City in their opening two Premier League games of the campaign.
Former Manchester United and Stoke City defender is still processing that Chelsea decided to sell one of their key players to a direct title rival.
“I was absolutely astounded that Chelsea allowed Nemanja Matic to go to Manchester United because he’s going to make them a better team,” Higginbotham told Sky Sports News on Friday.
“Manchester United are going to be challenging Chelsea for the Premier League. We’ve already seen the effect he’s had.”
Chelsea have also signed Manchester City goalkeeper Willy Caballero, AS Roma defender Antonio Rudiger and Real Madrid striker Alvaro Morata.
Manchester United are sitting at the top of the Premier League table following the additions of Benfica defender Victor Lindelof and Everton striker Romelu Lukaku this summer.
The Red Devils will host Leicester City in their next Premier League game at Old Trafford on Saturday evening.
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