The video which was posted on Snapchat back in February showed the older Zouma kicking the cat across his kitchen floor and slapping it in the face while Yoan filmed it.
The RSPCA have since taken custody of Kurt's two cats.
The animal welfare charity have now indicated that they will begin the prosecution of the Zouma brothers under the Animal Welfare Act.
An RSPCA statement said: "Following a full and thorough investigation, we have started the process of bringing a prosecution against Kurt Zouma and Yoan Zouma under the Animal Welfare Act.
"The two cats continue to be cared for by the RSPCA. We will be in a position to release more information once a court date is confirmed."
A West Ham statement said: "West Ham United is aware of the RSPCA statement in relation to its investigation involving Kurt Zouma. Kurt continues to co-operate fully, supported by the Club.
"It is our understanding that Kurt's cats have been checked by a vet, are in good health and have suffered no physical injuries. For legal reasons, neither Kurt or the Club will be making any further comment at this time."
National League side Dagenham and Redbridge, where Yoan plays, have issued their own statement, saying: "Dagenham Redbridge FC note the decision of the RSPCA to charge Yoan Zouma under the Animal Welfare Act 2006.
"Whilst Yoan continues to fully co-operate with the RSPCA, as this matter has now progressed further, Dagenham Redbridge FC feel it is no longer appropriate for either Yoan or the club to make any further statement.
"Dagenham Redbridge FC maintain their position in that the club continues to condemn all animal cruelty."
