No matter that his priority is securing a place in the top four, the Italian wants his side to lift the trophy. Next up is a trip to League One Notts County in the fourth round on Sunday and a chance for Mancini to get reacquainted with their manager Paul Ince.
"We want to win the FA Cup. We want to go to the final but first we must win this game," Mancini said.
"It is a very tricky match and the FA Cup is very tricky. When you play against a team like Notts County you think the game will be easy. But not me. They won at Sunderland. We will play on a small pitch and we must fight if we want to win."
Mancini knows all about fighting after being sent off playing for Sampdoria following a dust-up with Ince, then with Inter Milan. Sven Goran Eriksson was Mancini's coach and the City boss chuckled as he recalled the spat with the former Manchester United midfielder.
"I remember what happened, " said Mancini. "I had the ball and saw Paul Ince arrive (raising his voice) and I nutmegged him.
"It was incredible. Paul Ince was a good player with a good mentality. But I do not know that much about him as a manager."
In 2009 Eriksson was appointed director of football at Notts County and asked Mancini to become manager - an approach he rejected.
Mancini was unconvinced it was the right move and his instincts were proved correct as Eriksson left his post seven months later.
Mancini said: "Sven and I talked about this situation. It was a serious inquiry. When Sven called me for this job I read their history. But that was it."