Rooney is expected to inherit the captaincy from Nemanja Vidic next season.
Moyes says he sees the United captaincy as having a huge ambassadorial element to it, as well as leading the other players during matches.
"We want people to show leadership qualities, not just on the pitch but off the pitch. It is a big standing to be captain of Manchester United.
"The person who takes over has to understand what it means for the club, the responsibility it carries, what is required of you as well. I think we have a lot of leaders in the dressing room and they will all get a chance to stake their claim."
Asked specifically about Rooney's prospects, the United manager added: "Like everyone else, I'll give everyone that opportunity to stand up and show what they can do in the coming months."