The England international has been tipped to skipper the side when Nemanja Vidic leaves Old Trafford for Inter Milan in the summer.
While Patrice Evra currently holds the vice-captaincy, the Frenchman's future at the club remains up in the air, which could pave the way for Rooney.
Moyes has claimed that the 28-year-old was proud to stand in as captain against West Ham United on Saturday, in which he struck twice to secure victory at Upton Park.
"Wayne has led by example all season but maybe he took a wee bit of pride in wearing the armband to show exactly what it means," the Daily Mail quotes Moyes as saying.
"He wants to take responsibility. It's for him to show it and you have to say that he showed it. You get opportunities if you're a player out of the team who suddenly gets a chance or if you become a key player and then get the chance to wear the armband and take that on. He did that.
"I've said there will be no decision taken until next season. I thought Wayne picked up and did what he needed to do."
United, who are currently seventh in the Premier League table, will take on local rivals Manchester City at Old Trafford on Tuesday evening.