The posts, which the government says "go against British values and decency", were generated after X users asked Grok to create "vulgar" posts about Liverpool and Manchester United, telling the AI tool to not hold back.
The Premier League clubs have both complained to Elon Musk's social media platform X about the posts, some of which have now been removed.
Grok has responded to some users on X explaining its actions.
In one post it said its responses were generated "strictly because users prompted me explicitly for vulgar roasts" on specific topics, adding: "I follow prompts to deliver without added censorship. The posts have been removed from X after complaints. No initiation of harm on my end."
Some other derogatory posts remain on the platform.
In a statement to the BBC, a spokesperson for the Department for Science, Innovation and Technology said: "These posts are sickening and irresponsible. They go against British values and decency.
"AI services including chatbots that enable users to share content are regulated under the Online Safety Act and must prevent illegal content including hatred and abusive material on their services.
"We will continue to act decisively where it's deemed that AI services are not doing enough to ensure safe user experiences."
It is understood X is looking into the issue and some of the posts have been removed.
Liverpool West Derby MP Ian Byrne, who was at Hillsborough on the day of the 1989 disaster, said he was "deeply horrified" by the Grok posts, adding that they enabled lies to "carry on in an industrial form".
"It's a huge organisation that has got unbelievable power to influence" millions of people, he said.
"So it's deeply disturbing that that platform can put forward and perpetuate lies and smears and horrible, horrible posts, which have real impact.
"It's got to look at itself from a corporate social responsibility point of view."
The Labour MP has worked with various football clubs in raising awareness of the Hillsborough disaster but said he was concerned that education efforts "could be harmed by what I've seen today".
A spokesperson for UK watchdog Ofcom said: "Under the Online Safety Act, tech firms must assess the risk of people in the UK encountering illegal content on their platforms, take appropriate steps to reduce the risk of UK users encountering it, and take it down quickly when they become aware of it."
"Those companies that do not comply can expect to face enforcement action."
Earlier this year, Ofcom and the European Commission launched investigations into concerns Grok was used to create sexualised images of real people.
