Suarez is expected to receive a significant ban after appearing to bite Italy defender Giorgio Chiellini in Uruguay's 1-0 FIFA World Cup win on Tuesday.
Having previously bitten PSV Eindhoven's Otman Bakkal, while at Ajax in 2010, and Chelsea defender Branislav Ivanovic last year, Suarez's credibility is at an all-time low.
And Carragher – a former team-mate of the 27-year-old – slammed the Uruguay forward and believes his reputation has been forever tarnished, while he also does not think Suarez could change.
"Those who are saying the club (Liverpool) 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," Carragher said in his Daily Mail column.
"Now you get the feeling that Liverpool might actually be relieved if Barcelona or Real (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 will be remembered forever but that is no compliment. Rather than being regarded as one of the finest forwards in the game, he's now simply known as the one who bites people.
"I still find the incident in Natal completely bizarre. I worked with Luis every day for two-and-a-half years at Liverpool and if you spent time in his company at Melwood (the club's training ground) you would struggle to believe he was capable of such ridiculous moments.
"Away from that environment (at Liverpool), he has lost focus and if I'm honest, I don't believe he will change.
"He will always have that tendency to self-destruct. This latest incident mirrored the ones with Ivanovic and PSV Eindhoven's Otman Bakkal — it was a split-second reaction, instinctive."
Carragher also feels that Suarez will be significantly distressed by his actions but that was no excuse for his "disgraceful" actions.
The former England international does not believe Liverpool should be punished for Suarez's behaviour, though, and thinks that a one-year international ban is the appropriate course of action.
"Luis will be distraught," he added.
"When he bit Branislav Ivanovic during a 2-2 draw with Chelsea in April 2013, in the days and weeks that followed you could see how much of an impact it had made on him. He was devastated, as was his wife, Sofia.
"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' 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.
"I have thought about it and 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 in Chile next summer (in June and July next year).
"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?
"He missed the first five Premier League games, during which time Liverpool dropped five points against Swansea and Southampton.
"The form Luis was in last year, I'm certain he would have made the difference in those matches and Liverpool would have then won the title."
