Former goalkeeper Schmeichel won five league titles during his eight-year career with United, while City languished in the third division.
"In my first couple of years [at United] I didn't understand [the rivalry with City] - Liverpool and Leeds were the much bigger," Schmeichel told BBC Sport.
"Now it's different. Man City are the champions, they had their second Premier League win last season and maybe at the moment they are a bigger club than Manchester United in terms of results, so the significance has changed."
United have recorded three victories in nine league matches so far this season.