Former United goalkeeping coach Steele was axed by Moyes in the close season, along with assistant manager Mike Phelan and first-team coach Rene Meulensteen.
Moyes chose to bring Chris Woods, Steve Round and Jimmy Lumsden with him from Everton, whilst also appointing former United and Toffees midfielder Phil Neville in his first full-time coaching role.
United's 2-1 home defeat by West Brom on Saturday means the club has now made its worst start to a league season for 24 years.
With the pressure already on Moyes right from the start of his Old Trafford reign, Steele has now revealed how the summers changes were handled.
"He spoke to me, Mick and Rene," Steele is quoted as telling fanzine United We Stand. "I told him I thought it was a brave decision.
"He listened to the manager's advice, but he wanted to be his own man."
Steele arrived at United in 2008, having previously worked at Manchester City, and is credited with turning David De Gea into one of the league's best goalkeepers after a shaky start.
The 59-year-old says he respects Moyes' decision but is not convinced it was right for the club.
"I didn't want to leave. Why would I?" he added. "I knew that David was coming in and wondered who he'd bring.
"You had the United perspective - the manager saying, 'Keep what we've got, keep the continuity, work with them and they'll guide you through. You're taking on a massive machine here. You've gone from Marks and Spencer's to Harrods.'
"Then there's David's viewpoint, which I understand. I know him professionally. I know his work ethic, his hands-on approach.
"I don't blame him for doing what he did. I've been in the game long enough to know how it is.
"But it's ironic, isn't it? You've just been part of a team that has had a great season and won the league. David De Gea's had his best season. Does it make sense that you're not retained to continue the good work? Sadly, that's out of my hands."
United find themselves in 12th spot in the Premier League table after six games, eight points off the pace set by leaders Arsenal.
Despite the upheaval at Old Trafford, Steele still thinks the personnel in Moyes' squad will start producing better results.
"There were massive changes - the manager, chief executive, Scholes retiring and the coaches," said Steele.
"But there are enough players in that dressing room to win when it really starts.
"My time was up, I'm proud of what I did at United and I have great memories."