Moyes' former club are deep in crisis with supporters turning on Everton's hierarchy in the last few weeks as they face a second successive season battling to retain their Premier League status.
When Moyes arrived at West Ham for his first spell in 2019 one of his first matches saw angry fans invading the pitch and owners David Sullivan and David Gold forced to leave their seats amid toxic scenes.
But West Ham's fortunes have vastly improved over recent seasons with successive sixth and seventh-placed finishes and a Europa League semi-final last year.
Things have not gone to plan this season and Saturday's visit of Frank Lampard's side, who sit a place below the Hammers in the bottom three, could potentially cost one of the under-pressure bosses their job.
"It's a huge game for us both, yes," said Moyes. "But one you look forward to, I've got to. I've had a lot of big games and tomorrow is a big game in a different way. Yes, I'm looking forward to it, hopefully we can come out on top.
"I remember when I came here, I was only here three or four games, we had supporters running on the pitch. It was a terrible time for us and I think one of the biggest things that needs to happen as a football club is it for it to come together.
"That needs the supporters, that needs the players, the directors, all to be together. It's the one thing that happens here at West Ham, I don't see a divided club here and I've got to say from my time at Everton, I never saw a divided club at Everton.
"I can't really comment (on Everton now), I can only tell you that this football club is much more together I think than it's ever been."
Moyes also admitted he has been forced to go "off piste" with the signing of Danny Ings.
The West Ham boss turned to 30-year-old Ings from Aston Villa for around £12million in a bid to lift his goal-shy side with Michail Antonio and big-money summer signing Gianluca Scamacca failing to fire this term.
"I've had to go off piste a little bit because obviously we bought in players (last summer) and we wanted it to happen," he added.
"We tried to break it a little bit in the summer, to try to move it onto another level because I'd seen signs of it that we weren't quite going the way we wanted to.
"We've had to go and do things slightly differently because of the position we're in at the moment."
Ings is in line for an immediate debut against Everton with Scamacca sidelined for a couple of weeks with a knee injury.