Moyes believes that most of the Toffees fans that he shared a 'great relationship' with will appreciate the work he did at the club, but admits that it is likely that a section of them will greet his presence with boos.
"I wouldn't be surprised by football supporters, but I think Everton supporters know and are well aware of what was done," he 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 realise 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."
Everton are two points ahead of United in the Premier League table ahead of their clash tomorrow night.