In the defeat by Denmark, Maguire, who had had some tough months on and off the pitch, received his marching orders for two yellow cards in the first 31 minutes on Wednesday evening.
Keane says Southgate could have shown a softer side when a devastated Maguire walked past him down the tunnel.
Keane told ITV: “I think he sometimes [Gareth] knows that Harry had a hard time on and off the square, a pat on the back would have helped him.
“That would have set him off from Gareth.”
Keane also said returning to a subpar United side after losing two in their first three Premier League games won’t help Maguire’s form either.
“The only downside for Maguire is that he is returning to a team that is not full of confidence,” added Keane.
“I’ve been in a situation where I’ve been in a situation with brilliant characters. I felt like the dressing room is the best in the world. He won’t get that in these four years.
“I wouldn’t be around these guys – I don’t see them as leaders.”
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