The forward sunk his teeth into the biceps of Serbian defender Suárez during Liverpool's 2-2 Premier League draw with Chelsea on Sunday, prompting the club's managing director, Ian Ayre, to cancel a four-day trip to the far East and Australia where he was supposed to be promoting Liverpool's summer tour.
"I am deeply sorry for my inexcusable behaviour earlier today during our match against Chelsea," Suárez said.
"I have issued an apology and have tried to contact Branislav Ivanović to speak to him personally. I apologise also to my manager, playing colleagues and everyone at Liverpool Football Club for letting them down."
Later, he tweeted that he had been in touch with the Chelsea player. "I've just spoken to Ivanović on the phone and I could apologise directly to him," he said. "Thanks for accepting."
Ayre left Anfield before the incident to catch a flight to Asia but turned back from the airport.
"Luis [Suárez] has made an unreserved apology for his actions today," he said in a statement.
"His behaviour is not befitting of any player wearing a Liverpool shirt and Luis [Suárez] 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 Football Association."
Poor behaviour could cost
Manager Brendan Rodgers, who said he had not seen the incident, warned that poor behaviour could cost a player his place at a club.
"It doesn't matter who, players are always replaceable no matter how good they think they are," he said. "That is how football works. Of course, there are wonderful talents here we've seen at this club and others over many years.
"If you ever lose a player that you think you cannot replace, the next one still 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," he said.
"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."
On his return to Surrey, Ivanović was spoken to by Surrey police who wanted to see if he was injured and whether he wanted to make a formal complaint. He is understood to have decided not to press charges. – Sapa-dpa