Solskjaer is under pressure despite beating Everton on Saturday with the Red Devils having struggled in recent weeks.
Former Tottenham boss Mauricio Pochettino has been heavily linked with a job at United having been out of work for over a year.
And while Cole rates the Argentine, he insists that a new manager does not always equate to a change in fortunes.
“He [Pochettino] seems a very, very talented manager, just like the majority of managers at the highest level,” Cole told talkSPORT .
“But changing managers don’t guarantee anything, we have to be honest here, it doesn’t guarantee anything.
“Pochettino had an unbelievable spell at Tottenham, and still he wasn’t fortunate enough to win anything there.
“I don’t envy any manager. As soon as you take a job everyone is talking about how long have you got, the possibility of being sacked.
“Every week is a big week at Manchester United and every game is a big game, whatever manager is in charge they’re going to go through good and bad times.
“There is a lot of clamour at the moment from people who want to see Ole lose his job, but whoever is in the job, it doesn’t guarantee Manchester United success.
“Football is not as easy as that.”
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United came from behind at Goodison Park to win 3-1 with Bruno Fernandes netting a brace.
Solskjaer said: “If we can get this determination, mindset, approach every single game, we'll be hard to play against.
“You see all the tackles that Scott [McTominay] and Fred were winning in the middle there, we're tight, we're compact, we win the ball, we play, we've got spaces, that's how we want to see these boys. So we're very pleased with that.”