Van Nistelrooy, who served as caretaker for four games following Erik ten Hag’s dismissal, was keen to stay when Amorim arrived.
Now managing Leicester City, Van Nistelrooy will return to Old Trafford for their FA Cup fourth-round clash on Friday night.
"It was not a difficult situation," Amorim told reporters.
“Because it was so clear for me. I just had to explain. I think it is really simple.
"He understands, he is a man of football, he understands this situation. I have my team and my staff. I could keep Ruud because it was a good thing for me, a popular thing to do. But I will not put Ruud in front of my guys, no matter what, and then I had to put a person like Ruud in the end of the hierarchy on my staff, and I think that was not fair on Ruud.
"To show respect to Ruud, I was really clear with him. I explained I have my team and I will continue the same way and I will not put a guy like Ruud, a legend of this club, in the end of the hierarchy in the staff. So was the clear decision, really simple to explain and he took it like a class guy as he is.
"The most important thing to have as a coach is to have a staff who understand the way you want to play, the standards you have. And when I say standards I am not saying Ruud doesn't have these standards. He was a legend of this club, he knows what it takes to win, but I prefer to work with a team that brought me here.
"It’s clear that is important for me, to explain the same things to the players. I’m really comfortable and confident that I know how to win, no matter the club.
"Of course, we are losing and you need experienced guys. But I’m not lost, or I feel a (lack) of something in the staff. We just need to improve as a team."