David Moyes's men were humiliated by their title-chasing rivals in the reverse fixture in September when Manuel Pellegrini's men beat them 4-1 at the Etihad Stadium but, after two consecutive victories for his side, the England international is eager to avenge that defeat.
"We understand the fans and the result they want and for us as players it's exactly the same," he told Sportmasta.
"I've been part of this club for a long time now and you know what means a lot. This is certainly one of them.
United have also been thrashed more recently by Liverpool, who won 3-0 at Old Trafford earlier in March, and Carrick admits that the pain of that defeat still lingers in the dressing room.
"It was a bad day for us and we suffered just as much as the fans," he added. "It's horrible after a game like that. The dressing room was as you'd expect, really.
"We were very disappointed and frustrated. You don't watch the telly or leave the house for a couple of days if you can help it.
"You suffer from those results just like everybody else. Thankfully, we had a game to look forward to pretty quickly so that helped to give us something else to focus on but something like that, losing against Liverpool at home, hurts badly.
"We're well aware of what those games mean to everyone and, believe me, it means the same to the players but we can't go into Tuesday thinking about previous weeks."