The Sun published the video that was originally shared on Snapchat by Zouma's brother, Yoan, in which the France international can be seen picking up the family pet before kicking it across the kitchen floor.
Zouma issued an apology after the video emerged on Monday.
"I also want to say how deeply sorry I am to anyone who was upset by the video," Zouma said. "I would like to assure everyone that our two cats are perfectly fine and healthy.
"They are loved and cherished by our entire family, and this behaviour was an isolated incident that will not happen again."
West Ham also issued a statement promising that they will deal with the incident internally.
"West Ham United unreservedly condemns the actions of our player, Kurt Zouma, in the video that has circulated," the statement read.
"We have spoken to Kurt and will be dealing with the matter internally, but we would like to make it clear that we in no way condone cruelty towards animals."
The Royal Society for the Prevention of Cruelty to Animals was horrified.
"This is a very upsetting video," the RSPCA said. "It's never acceptable to kick, hit or slap an animal, for punishment or otherwise.
"We are so grateful to people who report suspected animal suffering to us and we would like to reassure people we will always look into and, if necessary, investigate any complaints made to us about animal welfare."
West Ham sponsors Experience Kissimmee confirmed that they would re-evaluate their relationship with the club after Zouma was named in Tuesday's line-up.
"It was disheartening to learn that the player Kurt Zouma was part of the starting line in West Ham United's game tonight," Experience Kissimmee tweeted. "As we wait for further information from WHU, we will be evaluating our relationship and sponsorship with the club."
Essex Police said on Tuesday they were aware of the video circulating on social media.
"We have recently been made aware that this incident may have taken place in Essex and we are liaising with the RSPCA and urgent enquiries are ongoing," a spokesperson said.