Suarez apologised first via his Twitter account and then through a statement issued on the club's website for an incident that came before he scored a dramatic equaliser seven minutes into stoppage time.
"I am deeply sorry for my inexcusable behaviour earlier today [Sunday] during our match against Chelsea," Suarez said in his statement.
"I have issued an apology and have tried to contact Branislav Ivanovic to speak to him personally. I apologise also to my manager, playing colleagues and everyone at Liverpool Football Club for letting them down."
Earlier on Twitter, Suarez had written: "I'm sad for what happened this afternoon, I apologize Ivanovic and all football world for my inexcusable behaviour. I'm so sorry about it!!" Suarez wrote.
The club have also responded to the incident, which came when the two players clashed following a challenge in the Chelsea penalty area.
The club's managing director Ian Ayre had cancelled a trip to Australia to deal with the fallout from the incident.
In a statement, he said: "Luis has made an unreserved apology for his actions today [Sunday].
"His behaviour is not befitting of any player wearing a Liverpool shirt and Luis is aware that he has let himself and everyone associated with the club down. We will deal with the matter internally and await any action from the FA."
Manager Brendan Rodgers had said in his post-match interviews that he had not seen the incident, but later said: "Having reviewed the video footage and spoken to Luis, his behaviour is unacceptable and I have made him aware of this."
Suarez looks set to face a Football Association investigation into the incident, which was not seen by referee Kevin Friend but later highlighted by numerous television replays.
It is not the first time Suarez has brought controversy to the club - he was banned for eight matches last season after being found guilty of racially abusing Patrice Evra - and he has previous history of biting.
Immediately prior to joining the Reds in January 2011 he served a seven-match ban while at Ajax after sinking his teeth into PSV Eindhoven midfielder Otman Bakkal.
Speaking after the match, Rodgers insisted no snap decisions would be taken on the future of the player, who is certain to attract interest from Europe's top clubs after scoring his 30th goal of the season.
"It is not for me to make any rash comments or any predictions now," he added.
"This is a football club where historically players treat people with how the football club respects society, players and everyone.
"There is certainly no one bigger than this football club, as a player or a manager.
"Players are always replaceable now matter how good they think they are. That is how football works.
"Of course there are wonderful talents here we've seen at this club and others over many years.
"You lose a player you think you can't replace but the next one comes along.
"The standards at this football club have been met for many years and that's why it is the worldwide institution that it is.
"The history of this club is about respect and how people are treated and that is something that will always be maintained here and will always be long after I am gone."
Rodgers said discussions would take place with the club's owners Fenway Sports Group - whose principal figure John Henry and chairman Tom Werner were in the city last week for the Hillsborough memorial service.
But he stressed players had to "accept the consequences" of their actions.
"I will always speak openly and honestly about the players and protect them when I can and if I think they are wrong I will tell them - as has already happened this season with Luis (when the striker admitted to diving).
"People have to accept it when they do wrong if that's what the case is.
"They have to accept the consequences accordingly. It's disappointing that we are not talking about the football."
Suarez scored the equaliser in the seventh minute of added time - only six extra had been indicated - having conceded the penalty with a handball which allowed Eden Hazard to put Chelsea ahead again after former Blue Daniel Sturridge had cancelled out Oscar's first-half opener.
Chelsea's interim manager Rafael Benitez, making his first return to Anfield having left in the summer of 2010, also did not see the biting incident and was more angry at the additional time played.
"It is a strange decision. Some games you have six substitutions and they say three or four minutes but six minutes (had expired) and with a throw-in for us the game was over," said the Spaniard, who enjoyed an emotional return with the Kop chanting his name numerous times.
"But six minutes and 45 seconds is difficult to explain. We need to analyse why we didn't stop the cross and they scored but in normal circumstances we would have finished the game."
Former Liverpool player and manager Graeme Souness labelled Suarez's actions as "embarrassing" and believes the Uruguayan is now in the "last chance saloon" as a Reds player.
"I'm not sure what to make of it really, embarrassing," he said in his punditry role on Sky Sports 1.
"A football club's board of directors' job is to attract and get the best football players and keep them at the football club.
"He is making it very difficult for himself to stay at Liverpool, I believe that puts him in the last chance saloon.
"More important than any of that is to safeguard the good name of the football club. This club is a world renowned football club. It is up there with any Barcelona, Real Madrid, Manchester United, it is up there with any of them."
Souness believes the incident came at a bad time for Liverpool, who this week marked the 24th anniversary of the Hillsborough disaster.
The 59-year-old pointed to Suarez's chequered past that has seen him involved in several controversies in the Barclays Premier League.
"People will be talking about this for a long, long time and it's going to show Liverpool in a very bad light - especially in this week of all weeks, anniversary of Hillsborough," he added.
"If you look at Suarez's track record in the past, his incident with Patrice Evra, this is not the first time he has bitten someone in a football match.
"It can't happen again.
"I firmly believe this puts him in the last chance saloon as a Liverpool player - the board have to see it that way because they're risking everything this great football club stands for."
Souness added: "Liverpool are not blessed with world-class footballers at this time, and he is genuinely a world-class player, but he's making it very difficult for them to hold on to him.
"I've never seen anything like that in a football match before. That's what children do when they are in the pram, they bite things if they are not happy with it.
"He must be on the borderline of doing something mad, crazy every time he goes out there if he is capable of that."
Former Liverpool midfielder Jamie Redknapp said what Suarez did was "indefensible".
He told Sky Sports 1 "He got a seven-game ban at Ajax for doing the same thing. He obviously has a real problem.
"There is that madness-genius gene in him because as a player, you hold your hands up - he's exceptional with people talking about him being player of the year, but what he did today [Sunday], is indefensible.
"Even the staunchest Liverpool supporters cannot look at that and think that's alright.
"You can't defend that and anyone who tries to is completely wrong, because the club is bigger than any player who has been on the football field.
"What's he doing? Why on earth would you want to take a chunk out of someone's arm when you are on a football field?
"That is an absolutely incredible act of brutality. It's madness."
Redknapp also believes Suarez should face a sizeable suspension.
He said: "I'd be surprised if he plays again this season. What message does will that send if he doesn't get a two or three-game ban.
"He's letting the club down."
Regarding the Uruguayan's long-term future at Anfield, Redknapp said: "He's giving the owners a problem.
"I wouldn't be surprised if he's still here (next season). I wouldn't think that's the end of him at Liverpool football club."
Professional Footballers' Association chief Gordon Taylor was disappointed Suarez's actions spoilt a good game of football.
While Taylor was coy over whether he would support a punishment, he expressed his disgust with the Liverpool forward.
"It was not good to see to say the least. It was unacceptable," he said on Sky Sports News.
"His reputation for controversy continues to increase when he can be such a good player, one of the world's best, so it is a real dilemma now.
"It's uncalled for, it's not good, it's depressing, it's such a shame for a player who's a really good footballer but this is not the first time that such things have happened.
"I know it's a really difficult one for Liverpool and the FA are going to have to deal with it.
"There's physical contact, emotions run high but no matter what players these days have to be considered as role models for youngsters.
"It was such a good game, great comeback, but this will take all the headlines."
Taylor admits his worry that Suarez is not learning from his controversial mistakes.
"It doesn't make it better when things have happened before and they're repeated again," he added.
"You begin to wonder, 'Is he going to learn his lesson?' Because when his football shines through he's so highly regarded so it's just a big shame."
Suarez later returned to Twitter to say he had spoken directly to Ivanovic.
"I've just spoken to Ivanovic on the phone and I could apologise directly to him. Thanks for accepting," he wrote.
Press Association Sport understands Suarez spoke personally with Ivanovic to apologise once Chelsea arrived back in the capital.
The Chelsea player is believed to have appreciated the call and he now considers the matter closed.
Ivanovic was also spoken to by Surrey Police at the club's Cobham training ground after complaints made to Merseyside Police were passed on to their southern colleagues, who wanted to ascertain whether the defender had sustained any injury.
That was found not to be the case and Ivanovic told police he did not want to make a complaint.
Chelsea also now consider the matter closed but will co-operate with any FA investigation should it be necessary.