The Spanish giants helped eliminate City at the group stages in last year's UEFA Champions League, but after exiting at the semi-final stage for the third consecutive season coach Jose Mourinho left the club to be replaced by Carlo Ancelotti.
Real also overhauled their squad by spending over €150m on a number of new recruits, most notably on Gareth Bale.
And Mancini believes those changes make the Real side the Turkish champions will face on Wednesday at the Bernabeu even more dangerous.
"I think that Madrid were strong last season and they are strong this year. You could even say they have improved because Bale has arrived and they have changed coach," Mancini said. "We are talking about the best of the best so to play here is always difficult.
"I think that Carlo is one of the best coaches in Europe. He has managed the best teams. He is a composed person, a good person, things that are very good in a coach."
Madrid are almost certain to go through to the knockout phase as they have a six-point lead in the group with just two games remaining.
However, Los Blancos will be significantly weakened by the absence of the competition's top scorer in Cristiano Ronaldo for Wednesday's encounter as he misses out with a slight muscular injury.
The Portuguese international's absence is a big boost for Galatasaray as they look to further their own chances of making the last 16, but they could be missing one of the own star men as Mancini admitted that Wesley Sneijder is not 100 per cent fit.
Galatasaray are currently level in second place with Copenhagen on four points, one point ahead of Serie A champions Juventus.
And Mancini is confident they can finish the job at home to Juve in two weeks time should matters still be in their own hands.
"If we don't do well tomorrow then Juventus and Copenhagen will have a better chance. We play the last game at home and we want to win it. Tomorrow will be difficult, but we will do what we can. We can beat Juventus."