The Reds have come in for criticism in recent weeks for the way they handled Luis Suarez's unsuccessful defence against allegations he racially abused Manchester United's Patrice Evra.
Last week there was also an incident in their FA Cup victory at home to Oldham when visiting defender Tom Adeyemi was allegedly racially abused by a fan, for which a man has since been arrested.
"Whatever the outcome of that investigation we are deeply sorry for what happened," Dalglish wrote in his programme notes for this Saturday's match at home to Stoke.
"We do not want and will not tolerate racism or discrimination anywhere near football and certainly not anywhere near this football club.
"The club is blessed with a worldwide fanbase, made up of different nationalities and diverse cultures, all of whom come together as part of the Liverpool family.
"The club will continue its proud record of fighting all forms of discrimination.
"Past, present or future, it has no place at Anfield, in our club or in the game.
"We will give any help we can to Tom, Oldham Athletic and the police to ensure this incident is dealt with properly."
Captain Steven Gerrard echoed the sentiments of his manager, adding in his programme notes: "Such incidents have no place in society, in football or at LFC.
"As a football club we have always supported anti-racism initiatives and our fans are well-known for their sporting nature and good humour.
"Any other type of behaviour will not be tolerated."