The striker said sorry on his personal Twitter account around three hours after the final whistle blew on the game at Anfield on Sunday.
The 26-year-old wrote: “I’m sad for what happened this afternoon, I apologise Ivanovic and all football world for my inexcusable behaviour. I’m so sorry about it!!”
In a statement issued through his club, he added: “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.”
The FA says it will review match footage – seen by millions of television viewers – that shows Suarez biting the Chelsea defender on the arm.
Cameras captured the striker appearing to sink his teeth into his opponent’s upper arm during the second half.
Ivanovic remonstrated with referee Kevin Friend following the incident and the match official briefly spoke to Suarez before allowing the game to continue.
The match ended 2-2, with Liverpool’s controversial star – who is a contender for this year’s player of the season awards – scoring an equaliser in the dying seconds.
In a later Twitter message, Suarez said he had spoken to Ivanovic and his apology had been accepted.
He wrote: “I’ve just spoken to Ivanovic on the phone and I could apologise directly to him. Thanks for accepting.”
A Chelsea spokesman said: “Luis Suarez called Branislav Ivanovic this evening. Branislav appreciated the call and his apology.”
Liverpool’s managing director Ian Ayre, who cancelled a trip to Australia to deal with the fallout from the incident, said: “Luis has made an unreserved apology for his actions today.
“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.”
The club’s manager, Brendan Rodgers, added: “Having reviewed the video footage and spoken to Luis, his behaviour is unacceptable and I have made him aware of this.”
Speaking on Sky Sports, former Liverpool manager Graeme Souness said the Uruguayan striker had “crossed a line”.
He said: “I’m not sure what to make of it. Embarrassing. He’s making it very difficult for himself to stay at Liverpool, that puts him in the last chance saloon.
“Most important is safeguarding the good name of the football club. Liverpool are world renowned. People will be talking about this for a long, long time, and it will show Liverpool in a very bad light.
“This is not the first time he has bitten someone in the football match. He is risking everything this great club stands for. I’ve never seen something like that in a football match before.
“There are lines you just can’t cross, and he has crossed it today.”
Professional Footballers’ Association chief Gordon Taylor was coy over whether he would support a punishment, but he expressed his dismay at the incident.
He told Sky Sports News: “It was not good to see to say the least. It was unacceptable. I know it’s a really difficult one for Liverpool and the FA are going to have to deal with it.”
But Marseille’s Joey Barton took to Twitter to urge Liverpool to back their player.
He wrote: “Suarez is a good as a player as there is in world football at the minute. Yes, he’s messed up and shouldn’t have bitten another player but a player like him has to play right on the edge.
“Without it he wouldn’t be the talent he is. LFC would be mad not to back him IMO. If they don’t I’m sure the queue of Champions League clubs would stretch round the block.”
Meanwhile, Suarez has found a new brother in arms in former heavyweight boxing champion Mike Tyson, who famously bit the ear of opponent Evander Holyfield.
According to Tyson’s profile page on Twitter, he made Suarez his 432nd and most recent follow.
Tyson took a chunk out of Holyfield’s right ear in the third round of a bout in Las Vegas in June 1997, after which Holyfield, nicknamed the ‘Real Deal’, became known as the ‘Meal Deal’.
Tyson was disqualified from the fight and fined \$3m (£1.9m).
Source: Yahoo Sports