The Italian manager believes if they can succeed this season it will have a snowball effect.
City face Aston Villa on Wednesday night in the FA Cup then Dynamo Kiev in the first leg of their Europa League last-16 tie next week in Ukraine.
Mancini is fully aware of the history of the club and the fact they last won a major trophy on February 28, 1976 when Newcastle were defeated 2-1 in the League Cup final.
He said: "We have a big chance of winning something. The next 20 days are very important because we play against Aston Villa then the following week in the Europa League.
"We have two options. If we win one trophy it will lead to more trophies. This is important for our team. When I accepted this job I knew it was difficult but it is a big challenge and I believe we can win a trophy."