A statement on the club’s website said: “We will not tolerate this behaviour being directed at any of our players and will fully support them in dealing with such incidents.
“Any action we can take unilaterally as a club will be taken immediately and we will also seek cooperation from other clubs where relevant, following that we will pursue the matter criminally with the full force of whatever the law allows.”
The club have not given any more details of when the abuse occurred.
Anti-discrimination campaigners Kick It Out said: “We fully support the response from Crystal Palace to the latest incident of vile racist abuse directed at Wilfried Zaha online.
“It’s extremely frustrating to continue to witness hatred on social platforms.
“While recognising it is a wider social issue, we utterly condemn the language used and demand that the platforms come forward with appropriate actions.”
Earlier this year, Zaha said in an interview he was subjected to racist abuse in “nearly every game” he plays in.
Last season the forward said on Instagram he was a target for racism, along with his family, after he won a late penalty in their 2-2 draw with Arsenal.
Zaha is one of a number of high-profile players to have suffered discrimination on social media.
In August, it was reported Twitter and Manchester United representatives would meet to discuss racist abuse aimed at Paul Pogba.
What is being done to tackle racism on social media?
Last month, football authorities met Twitter to discuss what can be done about the racist abuse of players on the social media platform.
The Premier League, EFL and the FA described the meeting as productive and positive.
In a joint statement at the time they said they examined “some of the specific issues around this unacceptable behaviour both online and offline”.