Liverpool say Luis Suarez misled them before refusing to shake Patrice Evra's hand, with the player and Kenny Dalglish having issued apologies.
Manchester United have thanked their rivals for the apologies and insist they want to "move on" from the whole affair.
Liverpool claim Suarez indicated to them he would shake hands with United defender Evra prior to Saturday's Premier League game at Old Trafford before refusing to do so on the pitch.
The Uruguayan, who was banned for eight matches after racially abusing Evra during the league game between the sides at Anfield in October, had been expected to shake hands with the Frenchman, but failed to do so.
On Sunday, Suarez apologised for his actions and Liverpool managing director Ian Ayre also released a statement criticising Suarez's actions.
Later, Reds manager Kenny Dalglish apologised for his conduct during his post-match interview with Sky Sports.
Ayre said: "We are extremely disappointed Luis Suarez did not shake hands with Patrice Evra before yesterday's game. The player had told us beforehand that he would, but then chose not to do so.
"He was wrong to mislead us and wrong not to offer his hand to Patrice Evra. He has not only let himself down, but also Kenny Dalglish, his team-mates and the club.
"It has been made absolutely clear to Luis Suarez that his behaviour was not acceptable.
"Luis Suarez has now apologised for his actions which was the right thing to do. However, all of us have a duty to behave in a responsible manner and we hope that he now understands what is expected of anyone representing Liverpool Football Club."
Suarez issued his own apology, which read: "I have spoken with the manager (Kenny Dalglish) since the game at Old Trafford and I realise I got things wrong.
"I've not only let him down, but also the club and what it stands for and I'm sorry. I made a mistake and I regret what happened.
"I should have shaken Patrice Evra's hand before the game and I want to apologise for my actions.
"I would like to put this whole issue behind me and concentrate on playing football."
Dalglish said in a statement: "Ian Ayre has made the club's position absolutely clear and it is right that Luis Suarez has now apologised for what happened at Old Trafford.
"To be honest, I was shocked to hear that the player had not shaken hands having been told earlier in the week that he would do.
"But as Ian said earlier, all of us have a responsibility to represent this Club in a fit and proper manner and that applies equally to me as Liverpool manager.
"When I went on TV after yesterday's game I hadn't seen what had happened, but I did not conduct myself in a way befitting of a Liverpool manager during that interview and I'd like to apologise for that."
United then issued their own statement, which read: "Manchester United thanks Liverpool for the apologies issued following Saturday's game.
"Everyone at Old Trafford wants to move on from this. The history of our two great clubs is one of success and rivalry unparalleled in British football.
"That should be the focus in the future of all those who love the clubs."
Suarez scored in a 2-1 defeat for Liverpool, with Evra appearing to taunt the South American by celebrating right next to him at the end of the match.
United boss Sir Alex Ferguson described Suarez as "a disgrace" and suggested he should never play for Liverpool again while Professional Footballers' Association chief executive Gordon Taylor branded Suarez's conduct as "disrespectful, inappropriate and embarrassing".
Taylor added: "If anything, Patrice Evra was the victim and he was prepared to put his hand out.
"These players are expected to be role models but if we have a situation where nobody accepts the findings of hearings and just carries on regardless, all you get is anarchy.
"Now the Football Association have to step in because the whole situation has gone too far.
"Suarez had a chance to put everything to be yesterday, in front of a worldwide audience. The fact that he chose not to is, quite frankly, depressing."
Having already defended his man in the flash TV interview with Sky Sports, Dalglish opted not to make himself available for further discussion on the matter.
On Saturday night, long-time friend Alan Hansen did not say on Match of the Day whether he had spoken to Dalglish prior to transmission.
However, his own observation was withering in its condemnation.
"Liverpool FC made a statement that there would be a handshake," he said.
"Luis Suarez was party to the statement. He knew about it and obviously agreed with it. To snub Patrice Evra was, in my view, totally unacceptable.
"The football club and the manager have given him total and unequivocal support through thick and thin.
"He has let Kenny Dalglish down, the club down and himself down."
what do you think?
Gerald Watson
Job done. Move on until another saga pops up.
ShellieWilliam1
I think the media need to move with this one also. Nothing was said when Rio Ferdinand refused to shake John Terrys hand a few weeks ago and RF isn't the victim or offender in that case. Also John Terry and Wayne bridge? Alex Ferguson needs to think back before critisising Kenny Dalglish for standing by his player that is a "disgrace" to Liverpool, did he not stand by Cantona after he launched himself studs first at a fan, to then keep playing him?? Luis Suarez said something he shouldn't of, and did admit to saying it but how many times are things said between players that are offensive whether it be about there race, country, family etc.
kyle rambo
Liverpool Fan? End of the day Show Racism The Red Card! Suarez Should of Been Sent Off Before The Game Started
Dorrien Phillips
@Kyle...yes closely followed by Ferdinand for doing exactly the same thing straight afterwards.
roger
this minor hiccup has been blown totally out of proportion and needs to be placed to rest, otherwise it will carry on and on, and is pointless. Things are sometimes done or said in the heat of the moment and not always meant. Q how mjany of us have done or said something we didnt mean or regret?
Brangane
The purpose of a handshake before a sporting event is a sign of sportmanship. Professional football lost much of its sportmanship when the money men got hold of it so maybe handshaking before a match is now a meaningless ritual anyway.
Gerald Watson
Whats the Cantona issue got to do with this???? Nothing. Does anyone remember the John Barnes & the Bananas thrown onto the pitch???? This whole thing has to be stamped out. Cantona was not involved in a race comment nor Was cloughie when he slapped a fan. Too many Man U haters ? Fine thats not a problem the problem is that A player was caught out Gross misconduct. Yes the club got behind him. Handled very badly & now he & the club are trying to put things right. (Fair Play) is here to stay.
Mark McNerlin
you are right cantons has nothing to do with race incidents but what people are pointing out quite rightly Fergie has no right to say to any football club to sack a player, what people are pointing out did any football manager ask for cantona or Ferdinand to be sacked after their incidents. yes suarez should have shook evras hand no excuse. but what would fergie have done if Ferdinand would have refused terrys hand the other week probably nothing
Michael Jordan
I am not saying that he is a racist, in terms of one who sets out to bring another person down, but that Suarez has not been properly prepared for the integration of the races in sport. What Suarez should have done is told Kenny (his boss) not to make a statement before the match. I think it would be a good idea if documents were prepared by the Players Union, in the Global Languages, that gave impartial advice on what Respect means in the Game of Football.
MarcPayne1
Horse, stable door, bolted.............................
Mike
It's about time we dropped this nonsence of formally lining up and shaking hands before a game. Do what rugby does and shake hands afterwards on an informal basis.
Gerald Watson
uturn.com springs ro mind. At least Spurs have nothing to do with it.
Chris Harvey
Well King Kenny, who's bang out of order now
pfmartin
Too many foreigners in English football, they are disrespectful, and don't follow their own club rules. Tevez, for one, just flits off as he pleases.Then we have the spitters, divers, cryers, liars,and generally poor sportsmanship all around., oh and this latest farce. In general, the foreign element have brought English football (the beautiful game ??), down to its knees.The good old days of hard tackling, and retaliation are long gone. It's all deception, and trying to get your oppo, in the refs eye. Typical foreign football. Garbage.
Gary Rimmer
pressure by the goverment nothing more needs to be said, lets fall in line with the torys forget the true fans and their thoughts
Alan Bell
Even the papers reporting this have no standards. Footballers, standards!! Suarez mind of a four year old. Kenny, oh dear! Alan Hansen fantastic. Let's move on.
Name witheld
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barry
Pity Liverpool as a club and others didnt educate their manager and players some acceptable manners and old fashioned Equity with so many players imported making up the teams, Kenny you have scambled egg all over you yes Bang out of order respect.
paddyj
Rio Ferdinand said that he would not have shaken John Terry's hand if he had been fit to play last week, but I supose that is ok, he also did not shake Suarez's hand, people presumed that was in retaliation for LS snubbing Evra, but he was halfway down the line shaking other players hands, looking them in the eye not at what was happening with the other two, so him and Wellbeck are just as guilty/petty as LS, the difference is they play for Man U and they arnt white and we wouldnt want to upset them now would we.
Craig Dobbin
2:55pm on 12/2/2012
I don't see the reason why he should have to shake his hand ... terry didn't have to
andrew hirst
9:45am on 13/2/2012
Terry wasn't the victim. If the victim holds his hand out and wants to forgive, forget and carry on, the perpetrator should respect that and not ignore it.