The Reds star drew controversy again on Tuesday when he appeared to bite Italy defender Giorgio Chiellini while on Uruguay duty at the World Cup, the third such incident of his career, and his old team-mate doubts that his "disgraceful" actions will end there.
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"I find the incident completely bizarre. I worked with Luis every day for two-and-a-half years at Liverpool and, if you spent time in his company, you would struggle to believe he was capable of such ridiculous moments.
"He is a special footballer because he scores incredible goals, [because of] his skill, tremendous physical capabilities and enthusiasm. He pushes himself to the limit but that doesn't mean we should have to accept moments of madness.
"Luis will be distraught. When he bit Branislav Ivanovic, in the days and weeks that followed you could see how much of an impact it had made on him. He was devastated.
"Do not think, however, I am looking to defend him. I didn't when he tangled with Ivanovic and I'm not changing my view. Luis's actions have been disgraceful and it defies belief that he could behave as if he was in a playground once again after everything Liverpool have done for him.
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Carragher is not in favour of a ban that would affect his club form after the incident took place on international duty but believes Liverpool are now likely to sell if any of the forward’s high-profile suitors were to make a move.
"I believe the biggest way to hurt him would have been for Fifa to suspend him for the rest of the World Cup and then ban him for a calendar year, which would rule him out of Uruguay's Copa America defence next summer," he opined.
"People are calling for a worldwide ban but I don't think Liverpool should be punished. Uruguay didn't suffer when he served the Ivanovic ban, did they?
"With Liverpool, his situation is even more complicated. Those who are saying the club should make a stand and drum him out have no grasp on reality because no business would get shot of an £80 million asset for nothing or a reduced fee.
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"Now you get the feeling that Liverpool might actually be relieved if Barcelona or Madrid come in with a bid that triggers a move. If a lucrative offer arrives in the next few weeks, I am convinced Suarez will be playing his football somewhere else next season.
"He has been a magnificent player and that is why Liverpool supporters have staunchly stood by him but even they believe now that this latest controversy is something the club can do without."