Cole, who played alongside Solskjaer under Sir Alex Ferguson, believes no manager deserves the "flak" the United boss has faced in recent weeks as it "doesn't sit well with me".
"The flak that Ole gets disappoints me," Cole told Sky Sports News. "I don't think any manager should take that kind of flak; whether it's United or non-league.
"I think they are trying to work to the best of their ability. He's trying like everybody else who tried previously. Whether he is my ex-team-mate or not, even if it was someone I didn't know, I wouldn't like people to talk about them like that.
"But I don't like it because when I played with him I got on very well with him. To see him in this position now doesn't sit well with me.
"We all want Manchester United to be back where they were, but that takes time. We can talk about 'how much longer?' But the previous managers have found themselves in the same position. The stick that Ole takes is, at times, not nice."
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