Moyes has admitted he'd like to see more new additions to his squad before the market shuts.
And Dunne told Tribalfootball.com: " I can understand his frustration because quite a few players have left Everton as well. They haven't had the strongest squad for several years now, they've become a team in and around the relegation spots in recent seasons.
"Moyes did brilliantly last season. He came in and dragged the club away from trouble quite early on. But they do need options, especially with the new stadium. You want to be going there watching a quality Everton side. He's already mentioned how late we are into pre-season, and the fact that he won't have what he wants yet for the start of the league season.
"If Everton want to recover and get themselves back into the top six or top eight, which is where people believe they should be, then they quite simply need players. These players will cost money, and it's not something Everton have been able to do in a long time.
"The positive for Everton is that it seems to be a big window this year. A lot of players seem to be moving clubs or looking for moves. At the moment, Everton are shy on squad numbers. They've probably got to wait and be patient for those targets to become a reality.
"The one thing with Moyes is that if he's given the players and given the backing, then Everton will eventually get back to where they want to be. There's a real possibility that Everton get back into Europe with Moyes, and he's already proved that at West Ham.
"For me, Everton are a bigger club than West Ham, so he can certainly give the club some good rewards if he's backed financially."
Branthwaite deal a big positive
One big success for Everton this summer, beyond their move to the new stadium, has been securing coveted defender Jarrad Branthwaite to a new contract.
And Dunne also said: "It's really positive for Everton. There have been some big clubs looking at him over the past couple of years, so to be able to tie him down tells me they've outlined the ambitions going forward to him. He'll be another player looking at the England squad for the World Cup, so for him not having speculation about his future is massive.
"It's important for Branthwaite that he stays fit. He needs to play the majority of the games this season, and take on that leadership role of being the main player in the defence. Jake O'Brien has come in and done really well, so you can potentially see a partnership developing there.
"Everton have some big centre-backs in there, so they will be a physical force. If Everton are going to improve, then they need to keep their best players, and Branthwaite fits that narrative. The one thing with centre-backs is, it's an ongoing battle to get better.
"Especially for a young player like Branthwaite. He's playing for a team that will be under the cosh at times, so he needs that ability to gain some control of the situation. If he can put his foot on the ball and relieve some of that pressure, then that will help the team massively because he has that ability to do so.
"If other teams see him popping the ball around them, then that stops the press to a certain extent. It'll be the mental side for Branthwaite, knowing when to slow things down, and I think you learn that more as you get older. He'll naturally improve as a player, as long as he stays fit."
- Richard Dunne was speaking to Tribalfootball on behalf of Betway