The Red Devils coach is still under pressure, despite his side getting an impressive 3-2 win in the Champions League at home against Atalanta in the group stages on Wednesday.
Solskjaer and United take on Liverpool at Old Trafford on Sunday in the Premier League.
"Absolutely, I think you have to [start dropping big players]," Johnson told PA in conjunction with William Hill.
"Mourinho had it when we were at Chelsea. The team he would leave out on a Saturday was pretty much good enough to finish in the top six.
"If you want to be the manager of a top club you have to be able to make the ruthless decisions. He has to make some big decisions and he has to leave some big names out. Mourinho literally couldn't care less.
"He knew he had a job and he protected his own job by picking a team to win a certain football match. If you want to work in big clubs that is what you have to do, you have to have that zero-tolerance attitude really.
"Ole comes across as a really nice guy and I can't imagine he wants to upset many people.
"That's fine, he can do it his way, he doesn't have to upset people, he can deliver the message however he wants. But he has to deliver it at some point."
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