The Scot left Goodison Park after 11 years to succeed Sir Alex Ferguson with the Red Devils, but he hopes the supporters remember his achievements at the club.
The Scot spent 11 years at Goodison Park as he retained the club's Premier League status and turned it into a regular challenger for a European spot, including a fourth-placed finish in 2005.
"I wouldn't be surprised by football supporters, but I think Everton supporters know and are well aware of what was done," Moyes told reporters. "The Premier League has been going just over 20 years, and they had a top-10 finish maybe once in that first 10-year period. In the second 10-year period, we finished in the top 10 on a regular basis.
"So I think if someone really looked at it, they would understand and hopefully realize I helped push the club in the right direction. I had a great rapport with the fans at Everton, but things wouldn’t surprise me. I’ve got to do a job for Manchester United, while Everton supporters have to support their team, which they’ve always done."
Moyes feels the fans should understand his desire to have taken over at United and admits he would still be at Everton if Sir Alex Ferguson had not retired.
"I don’t think there was any animosity built up from that, because I agreed a deal a month before the end of last season," Moyes added. "That was a decision taken by Manchester United. I had a contract and was deciding what my future was going to be at Everton and how we were going to go forward at Everton.
"In the meantime, four or five weeks before the end of the season, I got a call from Sir Alex. If I hadn’t received that call, I’m sure I’d still be working with (Everton chairman) Bill Kenwright as manager of Everton.
"That’s possible. It didn’t happen. For different reasons, it didn’t happen that way."