Albion have confirmed the matter has been reported to police.
Robinson posted a screenshot of the offensive messages he received on Instagram after the Baggies’ Premier League success at Stamford Bridge.
The 26-year-old had scored twice as relegation-threatened West Brom pulled off a stunning 5-2 win over the fourth-placed Londoners.
A statement from the club read: “West Bromwich Albion have contacted West Midlands Police to report racist social media posts directed at Callum Robinson following Saturday’s 5-2 win over Chelsea.
“The despicable Instagram posts were sent in reply to the Republic of Ireland international after he scored twice in a memorable team performance at Stamford Bridge.
“The club is appalled by the messages and will offer Callum the help and support he may require in response to the incident.
“Albion will continue to confront all forms of discrimination and will assist the authorities with their inquiries, seeking the toughest available legal punishment.
“It is deeply disturbing that scoring two goals in a football match can provoke a reaction of this kind. There is No Room For Racism. Anywhere. Challenge it, report it, change it.”
Robinson is the latest in a long line of footballers to be abused on social media, including Marcus Rashford, Reece James and Baggies team-mate Romaine Sawyers.
Chelsea later issued a statement in support of Robinson and said the club will take disciplinary action if any season ticket holder or member is found to have been involved.
“The club is disgusted with posts on social media this evening targeting Callum Robinson,” a Chelsea statement read.
“Chelsea FC finds all forms of discriminatory behaviour totally unacceptable.
“Racism has no place at this club and where there is sufficient evidence of a Chelsea season ticket holder or member involved in such behaviour, we will take the strongest possible action against them.”
The Premier League also added support for Robinson, urging all discrimination to be reported so that it can be actioned against.
A statement on Twitter read: “The Premier League stands alongside @WBA in condemning such abhorrent abuse, and discrimination in any form. There is #NoRoomForRacism, anywhere.
“Players, managers, coaches and their family members can receive support in cases like this through our reporting system for discriminatory abuse.”