The club confirmed a person was removed from the stadium and they are looking into the situation with urgency.
A United spokesperson said: "We are aware of an incident at yesterday's [Sunday] game where an individual in the home section has allegedly engaged in racial abuse.
"The individual was identified and ejected from the stadium. We are currently investigating the matter as a priority.
"Racism and all forms of discrimination are completely unacceptable and do not reflect the values of our club."
United did not comment on a potential punishment, though Liverpool will be in favour of "the strongest possible action" should someone be found guilty.
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The Reds also praised United's immediacy in beginning an investigation.
A spokesperson from the Anfield club said: "Liverpool Football Club takes all allegations of this manner extremely seriously and will support the strongest possible action being taken against anyone found to be responsible.
"At this stage it would be inappropriate to comment further on the alleged incident itself other than to welcome the swiftness with which Manchester United have launched an investigation and support their stance that racism and discrimination must be treated with zero tolerance."
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