But the new United manager brought his own staff with him from Everton and dumped Steele, Mike Phelan and Rene Meulensteen over the summer.
"He spoke to me, Mick and Rene," Steele told the United We Stand fanzine. "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."
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