Director of football Sven-Goran Eriksson was charged with finding the new boss and the Swede opted to turn to his countryman and former assistant at Manchester City and the Mexico national team Hans Backe, who signed a three-year contract on Tuesday.
Trembling and Eriksson discussed the job with Mancini but decided against taking matters further with the Italian due to the media interest his appointment would generate.
"Roberto Mancini was up there in our thinking, Sven and I met with him and he is a fantastic man," said Trembling in an interview in the Nottingham Evening Post.
"He is without doubt a fantastic coach, he could come into the club and do something from a PR point of view and he would put bums on seats.
"But I did not want a particularly high-profile appointment because we have had a media circus around the club for the last three months.
"It was important that we got somebody who would get on with the job and concentrate on the football side of things. I am sure Roberto would have wanted to do that, but I don't think the media circus would have allowed it."
Trembling is confident that Backe's experience - the Swede has won championship titles in Denmark and also managed in his homeland, Norway and Greece - makes him the right man for the job.
He also hopes the appointment will bring some stability to the club following a turbulent season so far at Meadow Lane.
"Hans is a highly regarded coach who is very well qualified and who has worked at a reasonably high level," said Trembling.
"He can also easily communicate with Sven, which is important, because he has such a great knowledge of football and it would be foolhardy not to tap into that.
"He has a presence in the dressing room, he can walk in and command respect and, at the same time, have the man-management skills to get the best out of his players.
"He is a good general communicator, somebody who, from a PR perspective, is good for the papers, for the TV and radio and presents the right image for the club. Ultimately, Hans was the right man for the job."
However, Backe has indicated this contract could be his last job in club management, and Trembling refused to rule out the possibility of considering Mancini again in the future when the club are closer towards their five-year goal of bringing Premier League football to Meadow Lane.
He added: "Well, maybe in the future. Hans is on a three-year contract and who knows what Roberto will do now? He has said publicly he is looking to get back into a coaching role.
"So who knows what he will be doing in three years time? If he is available in three years time and we are looking for a manager at that point, who knows?"