The former Manchester United striker returned to Old Trafford in the summer as part of Erik ten Hag's backroom team.
Van Nistelrooy later became interim manager after Ten Hag's exit in October, but followed him out the door after Amorim's arrival.
The Portuguese instead installed three of his former Sporting coaches as his assistants - Emanuel Ferro, Carlos Fernandes and Adelio Candido.
Speaking at his Leicester unveiling on Monday, Van Nistelrooy admitted he was 'hurt' and 'disappointed' not to be Amorim's assistant.
He told reporters: "The moment I took over the interim job what I said was I'm here to help United and to stay to help United, and I meant it.
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"So I was disappointed, very much so, and it hurt I had to leave.
"In the end I got my head round it because I also understand the new manager.
"I'm in football long enough and I've managed myself. I understand.
"I spoke to Ruben about it, fair enough to him, the conversation was grateful, man to man, person to person, manager to manager.
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"That helped [me] a lot to move on and straightaway get into talks with new possibilities which of course lifted my spirits."
Van Nistelrooy was confirmed as Steve Cooper's replacement at the King Power Stadium on Friday before watching the 4-1 defeat to Brentford from the stands the following day.
The role is his first permanent managerial position since leaving PSV last summer following a Dutch Cup triumph during his only season.
Van Nistelrooy takes over a side 16th in the Premier League and just a point above the relegation zone.
Leicester are also winless in four league matches, with just two victories to their name this season.
However, their new boss was unbeaten during his four-match United tenure.
Those matches included a pair of wins over his new club in the Carabao Cup and league.
Van Nistelrooy will take charge of the Foxes for the first time at home to West Ham on Tuesday night.