The Red Devils were keeping tabs on the 30-year-old star but decided against bidding for him.
Man United signed Rasmus Hojlund from Atalanta instead, but the Denmark international striker is not fully fit at the moment and could miss a couple of games.
The 20-year-old is also regarded as someone coming in for a long-term impact as opposed to one that would hit the ground running, and Redknapp feels the Red Devils missed a trick by not going after Kane.
“I’m still in complete shock about Harry Kane,” the former manager told the Sun.
“First of all, because I was so bullish that there was no way he would go to Bayern Munich.
“But also because I cannot believe that someone in the Premier League, especially Manchester United, didn’t go for him.
“If Man United have got ambitions of winning the Premier League, he’s the one who could have made the difference for them.
“I really think he could have tipped the balance for United.”
Kane had just 12 months left on his deal at Spurs and was not interested in signing an extension, forcing the club to sell him instead of losing him for free next summer.
Having such a player leading the line for Man United would undoubtedly have been huge.
However, Tottenham chairman Daniel Levy would not have sold the striker to another English club.
He would not have been eager to strengthen a direct rival, and the Red Devils cannot be blamed for not bothering to open talks with Spurs for Kane.
They will hope Hojlund can produce the goods given his massive potential, and the youngster will hope to become available sooner rather than later as he continues to train individually ahead of his debut.
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