The Italian replaced Fatih Terim as Galatasaray boss last week to secure a return to management following his departure from Manchester City at the end of last season.
He picked up a 2-2 draw away to Juventus in the UEFA Champions League in his first game last week, but lost 2-1 at Akhisar Belediye in the Turkish league at the weekend.
Mancini told Gazzetta dello Sport: "In Italy, we always think about money. It's clear that contracts are part of our profession, and they have importance, but I accepted their proposal because Galatasaray are taking part in the Champions League and they have a relevant history.
"I'm fascinated by the idea of facing a new experience in a country that is a bridge between Europe and Asia."
Galatasaray already trail Fenerbahce by nine points in the domestic table and Mancini is hoping his new club can make progress in Europe.
He added: "The draw we got in Turin in the Champions League gives us a little more hope.
"In our league, we are a little late, but we can recover. The other big sides, and I'm referring to Besiktas and Fenerbahce, have the advantage of being able to concentrate only on the league because they are out of the European tournaments.
"Will we strengthen in the winter? Let's see what happens in the Champions League.
"If we go forward, something will be done, otherwise we could remain as we are. It's up to the club. In the last few years, some important investments have been made."