Television images circulating on social media showed a City fan making a gesture in front of United's Brazilian midfielder Fred during the game, which the visitors won 2-1.
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Fans also threw objects on to the field when Fred was taking a corner in front of them and he was struck at least once.
"A 41-year-old man has been arrested on suspicion of a racially aggravated public order offence and remains in custody for questioning," police said in a statement to the Daily Mirror.
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Superintendent Chris Hill of the City of Manchester Division added in the statement:
Racism of any kind has no place in football or our society and I hope this arrest shows that we are taking this matter extremely seriously.
City said on Saturday that they had a zero tolerance policy regarding discrimination of any kind and anyone found guilty of racial abuse will be banned from the club for life.
The Football Association is also investigating the incident.