Moyes has begun work as Manchester United manager after Everton's 2-1 loss at Chelsea in his 518th and last match in charge, while Neville is taking time to consider his future following the end of his playing spell at Goodison Park, eight years after his arrival from Old Trafford.
Moyes anticipates the 36-year-old Neville's next move will be to stop playing altogether, saying: "I think he'll probably go into coaching or management, that would be my tip."
He added: "He has got a bit of a problem with his knee, but he'll know if he can play on or not. He's a great man and has a lot of good values. I can see him being successful in the future."
On whether Neville could be a success at Everton, who are now looking for a new manager after the end of Moyes' 11-year reign, the Scot said: "He could (stay on). He's capable, he's got good values, he knows the job and knows what's expected. He could do that."
Moyes will be happy to share his thoughts with Everton chairman Bill Kenwright and the board if asked about his successor.
"Bill and the board will be the ones who make the decision," Moyes said. "There's a pretty good structure in place. There are some restrictions at Everton, but there's a good group of players.
"It's a decision: do they follow the same path or will they maybe try something else? That'll be down to the chairman and the board of directors. There have been a lot of things I've hopefully improved. I'm really pleased with how the football club has moved forward on the field and hopefully it continues to do so."
Steven Naismith, who scored Everton's goal at Stamford Bridge on Sunday, joined from Rangers last summer. He was among those to pay tribute to Moyes following the manager's final match in charge.
Naismith told Everton TV: "It's been strange, because there's been a lot of speeches made about the manager and what a job he's done. Every word that could be said about him has been. He moved this club from a relegation fight to a team that's challenging for Europe every year. It's full credit to him."
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