A 27-year-old man and a 17-year-old boy, both from Stoke-on-Trent, were questioned following the Potters' clash with Manchester City at the Etihad stadium on Saturday.
The man was released on bail until a date in March and has been banned from entering Greater Manchester Police force's area in the meantime.
The teenager was handed a police caution.
A spokesman for Stoke City said: "As a club we abhor discrimination in any form and fully support Greater Manchester Police and Manchester City in this matter."