The alleged incident occurred in the second half of City's 2-1 defeat at the Etihad as a supporter appeared to make a monkey gesture towards at least one United player.
In a statement, Greater Manchester Police said: "A man has been arrested after a video posted on social media showed a supporter making racist gestures at the Manchester City v Manchester United match at the Etihad Stadium yesterday".
"The club are also working with GMP regarding objects being thrown on to the field of play".
A video circulating on social media after the match appeared to show a supporter aiming racist abuse at United midfielder Fred and the Brazil global told ESPN Brasil he blamed the "backward society" for the incident. The Brazilian midfielder spoke to the referee about the abuse and attacks. I saw it only in the locker room afterwards. He even threw a lighter and it hit me.
United manager Ole Gunnar Solskjaer branded the alleged abuse "unacceptable".
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"Officials from the Club are working with Greater Manchester Police in order to help them identify any individuals concerned and assist with their enquiries.
The club operates a zero tolerance policy regarding discrimination of any kind and anyone found guilty of racial abuse will be banned from the club for life".
"We will continue to work with Manchester City and Manchester United Football clubs on this incident and will investigate any other lines of inquiries".
Speaking to ESPN afterwards, Fred said: "Unfortunately, we are still in a backward society".
Before the start of the match, City were the all-out favourites to win this game and keep their hunt for the league title alive but surprisingly it was the away side who started on the front foot and went ahead 2-0 in rather quick time.