Greater Manchester Police said a 29-year-old man was in custody on suspicion of racially aggravated assault and the incident, involving City forward Raheem Sterling, was being treated as a hate crime.
"Racism will not be tolerated in Greater Manchester and has no place in civilised society," Detective Chief Inspector Paul Walker said in a statement.
"We take every report incredibly seriously."
Sterling scored twice in City's 4-1 defeat of Tottenham Hotspur at the Etihad Stadium on Saturday evening.
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The club has yet to comment.
(Reporting by Martyn Herman; Editing by Richard Balmforth)
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