However, the closure of the English transfer window means United will not be able to sign a potential replacement for the Frenchman.
He has been linked with a move to Real Madrid as well as a return to Juventus, who the Red Devils bought Pogba off for £86million in 2016.
United were busy in the transfer window, spending £145million on new players including defenders Harry Maguire and Aaron Wan-Bissaka as well as Welsh midfielder Daniel James.
Pogba has reiterated his desire to ‘give everything’ while at the club and he helped United get their Premier League campaign off to a flyer with a 4-0 victory over Chelsea.
And McClaren believes Pogba still has plenty to give.
McClaren told Sky Sports News: “I think he will stay, he would have gone by now. They would have replaced him and now they don’t need to, that’s the key thing.
“You could see it on Sunday, Chelsea did really well against him in the first half, but once he got space in the second half, he can deliver the pass, score the goal. He can do things.
“When Ole first came in, he got the best out of Pogba for the first 15 games and the way he turned the game around against Manchester City, that’s what he’s capable of doing.
“Ole believes he can get the best out of him and especially with Maguire and the way they have started, Pogba might think ‘yeah, I could be the king of this team’.
“These players want top players around them, to be challenging for the trophies. Manchester United demonstrated that with the summer signings [and] the first game.
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“Pogba might be thinking ‘they really want me and this is a good team, I could win things here’.”
United new boy Maguire was praised for his performance against Chelsea with former Red Devils boss Jose Mourinho saying the defender was his man of the match.
And McClaren claimed the former Leicester City man could have a similar impact at Old Trafford to what Virgil van Dijk has had at Liverpool.
McClaren added: “[They have] signed [Aaron] Wan-Bissaka and the young boy from Swansea [Daniel] James. But the biggest of the summer was Maguire.
“He’s like van Dijk signing for Liverpool, the difference – the person that took Liverpool to the next level. I think Maguire can do exactly the same for Manchester United.
“He did it on Sunday against Chelsea, he was magnificent. He’s a great defender, a leader.
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“The way he speaks to the players and his demeanour, confidence and courage when he has the ball, I think he’s the perfect signing for Manchester United.”
Liverpool only made three signings this summer with goalkeeper Adrian, defender Sepp van den Berg, 17 and forward Harvey Elliott, 16 arriving.
And McClaren suggested Reds boss Jurgen Klopp could come to regret this lack of action in the transfer window.
McClaren said: “Sir Alex Ferguson always said at United, at the end of every season we’ve won something, there’s going to be one or two coming into that dressing room that makes the players sit up and go wow, we’re going onto another level.
“That’s the only thing that worries me. They didn’t go on and get another star striker and players are now playing around the clock 12 months of the year.”