The Hammers will travel to Goodison Park on Sunday, with former Everton boss Moyes looking to improve on his record of two wins from 15 Premier League matches against sides managed by Benitez.
West Ham are currently ninth in the table with three wins from their opening seven matches, while Everton are three points better off in fifth, level on 14 points with Manchester United and Manchester City.
“I can speak because I’m the manager of West Ham and my plan is to challenge the top four,” Moyes said.
“I’m hoping I can challenge the top four and maybe catch one, maybe topple one. That’s what I want to do. That’s what I’ve let the players know in terms of where I’m aiming.
“I can only speak for Everton as ex-manager. I can’t speak for Rafa. But Everton is a big club, with big traditions and they’ve challenged the top four, certainly in my time there.
“There was a constant challenge. It was really difficult to break in, but we did it on one occasion. So I think Rafa might be thinking the exact same as I did at the time, he may be thinking even more.”
The pair met a number of times in Merseyside derbies when Benitez was in charge of Liverpool and Moyes managed Everton, and the West Ham boss was complimentary about the new Toffees chief.
“I think there is an identity to Rafa, I think his teams are really well-organised, well-coached and I think he prepares his teams in a fashion where I think he understands the level of players that he has got,” the West Ham manager said.
“If he has an extremely talented group of players he can play a more expansive style, if he has a group he thinks are maybe not quite so strong he can play more tighter and maybe not open up quite so much.
“But that comes from being a good coach and a good manager and understanding the group of players you have got.”