The Uruguay international has been charged with violent conduct by the Football Association after the incident in Sunday's 2-2 Premier League draw with Chelsea at Anfield.
Suarez's actions against Ivanovic, for which he apologised on Twitter on Sunday evening, have caused mass controversy and he is facing up to a ban of more than three matches.
Given Suarez was last season suspended for eight games after being found guilty of racially abusing Manchester United's Patrice Evra, while he was also banned for seven matches when playing for Ajax after biting PSV Eindhoven midfielder Otman Bakkal, there have been calls for Liverpool to sell their star man.
The Merseyside club have, however, said they have no plans to show Suarez the door and they have instead fined the South American while the Professional Footballers' Association have offered the player help with anger management.
And former Manchester United defender Gary Neville, who was in 1995 a squad-mate of Eric Cantona at the time when the Frenchman was banned for nine months by his club for kicking a spectator in a match at Crystal Palace, does not think Liverpool should ever have considered getting rid of Suarez.
Neville admitted he was 'shocked' by Suarez biting Ivanovic but he also said on Monday Night Football: "You hear all this media frenzy, 'get him out of the country'. The only way I can try and compute this, forget being a pundit and analysis when I am sat here in this chair, is to put myself back in the changing room.
"As I have had team-mates who have done wrong and as I have done wrong myself in the past, do I throw you overboard? I would think they have let their team-mates and their football club down and created pressure.
"But I would never for one minute think, 'throw the man overboard'. The idea Suarez can never play for Liverpool again or the idea he should be thrown out of the country, for me, is nonsense and an overreaction.
"He has made a big mistake. He knows that. He has got to try and win people back. He has got to get better. When you make mistakes, you have got to get better."
Liverpool were last season criticised for their slow reaction to the controversy surrounding Suarez after his racial abuse of Evra but the club have acted quickly following this weekend's incident.
And Neville believes the Anfield outfit have improved on the mistakes of the past, with managing director Ian Ayre even postponing a trip to Australia in order to oversee proceedings.
Neville said: "They caused themselves problems last year with the protracted incident around Patrice Evra. This time, they have dealt with it quickly and swiftly.
"I thought Liverpool and Brendan Rodgers dealt with it very well after the match in a difficult position. Luis Suarez has also done all he can to defuse and make the situation better for him.
"From the mistake onwards, Liverpool, Luis Suarez and the manager have done all they possibly can to try and make it as best a situation as they can."
Liverpool and Suarez have until 6pm on Tuesday to respond to the FA charge of violent conduct before they learn the length of any ban.
The FA believes a three-match suspension will not be sufficient but Neville does not want to speculate over the possible punishment.
He said: "If you are asking me for a number, I cannot give you any. You could throw a dart at a dartboard. Luis Suarez will accept the charge, he will get on with it.
"I would be surprised if he plays again this season. He has just got to take his punishment. He has accepted he has done something wrong. It is a mistake."