Zaha posted on Twitter three screenshots of messages containing racial abuse from an Instagram account belonging to a Villa fan.
"We deplore the disgusting racist messages sent to @wilfriedzaha," Villa said on the club's Twitter account.
"We condemn all forms of racial discrimination and stand with @CPFC . We are working with the police in investigating this extremely serious matter and when the culprit is identified AVFC will issue a lifetime ban."
All Premier League games since the season's restart last month have begun with players, staff and match officials taking a knee to protest against racial injustice.
The shirts of all 20 sides in the English top-flight have also carried a logo promoting the Black Lives Matter campaign.
"This is an absolute disgrace and should not be happening. We stand with you, Wilf, and anyone else who has to suffer such horrific abuse," Palace said on Twitter.