The Red Devils go up against Fulham in the fourth round of the competition this Saturday as they try to clinch the trophy that they have not won for eight years.
"It's definitely a trophy I want to win," Cleverley told the club's official website. "It means a lot. In fact, we've got a few players in the squad who haven't won anything [major] in this country as yet, including myself, so anything we're involved in I want to win."
The North West side are on course to snatch the Premier League title from rivals Manchester City this season as they sit five points clear at the top of the table.