The 28-year-old has emerged as a key member of Ole Gunnar Solskjaer’s squad this season.
Fred has made 38 appearances in all competitions, playing in both central midfield and in defensive midfield.
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He has laid on one assist but has not scored a goal, and has played in all of the club’s last 10 games.
Gillespie, though, believes that he lacks the ability to pass the ball effectively and has called on United to try to invest in their midfield.
Indeed, he thinks that Nemanja Matic is ageing and has his doubts over the quality of Scotland international Scott McTominay.
Speaking exclusively to TIF, he said: “You’ve got Matic who is coming towards the end. You’ve got Fred who, for all his hard work and endeavour, just isn’t Man United class for me.
“He gives the ball away too often, you just always feel like there’s a mistake in him.
“And you’ve got McTominay who at times looks the part and then other times doesn’t.”