The player issued his own apology - something he failed to do adequately when found guilty of racially abusing Manchester United's Patrice Evra last season, for which he was banned for eight matches - first on Twitter and then via an official club statement.
"I am deeply sorry for my inexcusable behaviour earlier today during our match against Chelsea," he said.
"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."
Later in the evening, Suarez returned to Twitter to say he had spoken personally to Ivanovic.
"I've just spoken to Ivanovic on the phone and I could apologise directly to him. Thanks for accepting," he wrote.
Ayre, having been one of many criticised for a lack of leadership during the Evra affair, took personal charge of the situation and held a meeting with Rodgers after the match in which they reviewed footage and formulated their response.
"Luis has made an unreserved apology for his actions," said Ayre.
"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."
Referee Kevin Friend did not see the incident but television pictures are conclusive and the FA are almost certain to act.
Suarez has previous in relation to biting as, 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.
That will have no bearing on what action the FA take, however, as it was under a different association.
Rodgers insists no snap decisions will 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.
"There is certainly no one bigger than this football club, as a player or a manager.
"Players are always replaceable no matter how good they think they are. That is how football works.
"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."
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 header.
Chelsea's interim manager Rafael Benitez, making his first return to Anfield having left in the summer of 2010, was angry at the additional time played.
"It is a strange decision. 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."
Suarez spoke personally with Ivanovic to apologise once Chelsea arrived back in the capital.
"Luis Suarez called Branislav this evening. Branislav appreciated the call and his apology," said a Chelsea spokesman.
Press Association Sport understands 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 press charges.
The matter will now most likely be dealt with by the Football Association, and Chelsea and Ivanovic will assist with any investigation should it be necessary.
