Holgate pushed Brazilian Firmino into the advertising hoardings in the 40th minute of the third round tie and the Liverpool forward reacted by verbally abusing the young Everton defender.
"The club and player will fully co-operate with the relevant authorities to ensure the facts are established in a thorough manner, if deemed necessary or requested. While that process is ongoing, we will not be making any further comment," Liverpool said.
Holgate, who is mixed-race, complained forcefully to the referee Bobby Madley about Firmino's comments and there was later a conversation between the referee and the fourth official on the touchline.
Asked by a reporter whether Holgate had accused Firmino of using a racist term, Everton manager Sam Allardyce said: "I'm telling you nothing until whatever systems are used to find out what did and didn't happen. I'm here to talk about football, not controversial incidents. They have to be dealt with by the authorities."
Allardyce struck a similar tone when asked whether he had discussed the matter with Holgate.
"Even if I had or I hadn't, I wouldn't tell anyone in the press conference. That is for other areas of the football club. I have told the Director of Football (Steve Walsh) to sort that out."
The incident will almost certainly feature in the match report submitted by the referee to the Football Association who would then make any decision on eventual disciplinary proceedings.
Liverpool won the game 2-1.
(Reporting by Simon Evans, editing by Pritha Sarkar)
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