Everton manager Sam Allardyce and Liverpool counterpart Jurgen Klopp refused to be drawn on the incident between Roberto Firmino and Mason Holgate.
Firmino and Holgate were involved in an altercation during Liverpool's 2-1 FA Cup derby win at Anfield Friday.
Holgate pushed Firmino, who was shielding the ball over the sideline, into the crowd before the Liverpool forward reacted angrily.
A Liverpool spokesperson said: "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."
The team's respective managers were not drawn on the incident, with Allardyce saying he had left it in the hands of director of football Steve Walsh.
"I'm telling you nothing until whatever systems are used to find out exactly what happened and what didn't happen," he said.
"I'm here to talk about football, not about controversial incidents. I only want to talk about football.
"Anything else outside of that that happens has to be dealt with by the authorities."
Klopp was also unwilling to reveal what happened between the pair, saying he initially thought Holgate had committed a foul.
"I only heard about something but I can't say anything about it because so far I couldn't speak to anybody," the German said.
"I think from now on, first of all the club will answer. It's like this, I don't really know anything about it.
"I misjudged the situation obviously completely because I thought it was about the Holgate foul. At the end, there was no foul, no yellow card, nothing. I didn't get it to be honest.
"It was a throw in so I didn't understand the whole situation after the game and I got information that something happened, but I can't say anything about it."
Virgil van Dijk scored the winner on his debut to send Liverpool into the FA Cup fourth round.
