The Turkish champions sit joint second in Group B with Copenhagen, some six points behind runaway leaders Real Madrid with two games to play.
Juventus are a point further back, and Galatasaray's next match sees them take the daunting trip to the Bernabeu.
However, despite the close nature of the pool, Mancini - who took over at the Turk Telecom Arena in September - is hopeful his side can secure their place in the first knock-out round.
"Galatasaray are a very good team who have the potential to do well this season. It's never easy in the Champions League and our group is particularly tough. However, we can progress," the Italian told UEFA.com.
"I'm still confident of qualifying. We need to improve as a team but I am hopeful of a good result in our next game against Real Madrid.
"Then we will have everything to play for (at home to Juventus on 10 December). As they say in football, never say never."
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