The Uruguay striker appeared to bite the Chelsea defender's arm midway through the second half of the 2-2 draw at Anfield before heading in an equaliser deep in stoppage time.
And given Suarez's history of courting controversy during his spell with Liverpool, Souness believes this latest incident could prove to be the final straw for the club's board.
"I'm not sure what to make of it really, embarrassing," he told Sky Sports following the match.
"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 Barcelona, Real Madrid, Manchester United, it is up there with any of them.
"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, the anniversary of Hillsborough.
"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.
"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."
Former Liverpool midfielder Jamie Redknapp said what Suarez did was "indefensible".
He said: "He got a seven-game ban at Ajax for doing the same thing. He obviously has a real problem.
"Even the staunchest Liverpool supporters cannot look at that and think that's alright. You can't defend that and anyone who tries to is completely wrong."
Redknapp also believes Suarez should face a sizeable suspension, adding: "I'd be surprised if he plays again this season. What message does will that send if he doesn't get a two or three-game ban.
"He's letting the club down."
Regarding the Uruguayan's long-term future at Anfield, Redknapp said: "He's giving the owners a problem.
"It's such a shame because he's playing for such a great football club. How many more chances are you going to give him?
"I wouldn't be surprised if he's still here (next season). I wouldn't think that's the end of him at Liverpool football club.
"It's not the first time now but footballers are valuable commodities, they certainly aren't going to sack him for that, the only thing maybe is they will sell him in the summer as they think it's one time too many now."