In the wake of the Reds’ 1-1 draw with Manchester United last Saturday, Patrice Evra claimed Suarez had verbally abused him on a number of occasions during the game.
Suarez refuted the allegations, which the FA are looking into.
But Dalglish, wearing a badge promoting anti-racism organisation Kick It Out, fully backed the player.
He also rejected suggestions by United counterpart Alex Ferguson, made after the game, that the Uruguay international was a diver.
“The only thing I will say — and then it is put to bed — is that the football club and everyone at the football club is totally and utterly fully behind Luis Suarez,” said the Scot.
“There was an allegation after the game about him diving all over the place and there was an allegation from Patrice Evra.
“The two things are emotive but we support the wee man on both things.
“At the start of the week the club made their position known and apart from what I have said we won’t be adding to it.
“We are fully looking forward to a complete, transparent report from the FA and we will co-operate 100%.”
So far there appears to be little progress on the matter, with the FA due to interview Evra yesterday.
The France international is likely to have to provide evidence in the form of witnesses as he did not raise the allegations with referee Andre Marriner at the time.
The official only received a complaint after the game when Evra, accompanied by manager Alex Ferguson, asked him to put it in his match report.
Suarez went public with his denial on Sunday and until corroborating evidence can be found it is one player’s word against the other.
Other matters off the pitch have dominated this week after the British government pledged to release all documents relating to the 1989 Hillsborough disaster, in which 96 Liverpool fans died, to the independent panel overseeing which papers should be made public.
Yesterday retired judge Oliver Popplewell, who conducted the official inquiry into the Bradford fire, provoked outrage when he criticised Hillsborough campaigners.
Popplewell said in a letter to The Times that they were “harbouring conspiracy theories” rather than behaving with “quiet dignity and great courage” like the Bradford families did. Dalglish said he was not aware of those comments but offered a dismissive response.
“Who is he? Why should he make someone else’s mind up?” said the Reds boss, who was in his first stint as manager at the time of the 1989 disaster and was so personally affected that it contributed to him quitting less than two years later.
“If people want to find solace in something then fine. I don’t think he should be interfering. I don’t know who the guy is or what he said but certainly the Hillsborough families have been hugely complimented on their dignity and how they have gone about what they want to achieve.
“Our families have been fantastically dignified in their approach and also, after 22 years, very patient and the least they deserve is some comfort.”