The controversial forward once more attracted the headlines for all the wrong reasons after appearing to bite Ivanovic midway through the second half of Liverpool's clash with Chelsea.
That the 26-year-old scored a late leveller to ensure the Merseyside club escaped with a 2-2 draw against the Champions League chasers at Anfield is very much an afterthought.
Instead, much of the post-match reaction has centred on his aggression on Ivanovic, with Souness stunned by the incident.
"I'm not sure what to make of it really, embarrassing," the former Liverpool player and manager 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."
It is not the first time Suarez has been involved in such an incident.
In 2010 he was handed a seven-match ban by the Dutch FA and fined by his club for biting PSV Eindhoven midfielder Otman Bakkal during an Eredivisie match.
It would be his last appearance for Ajax as he joined Liverpool two months later.