The Uruguay striker has spoken of his desire to join the Spanish giants this summer and leave behind the club that stood by him through a number of controversies, including the Patrice Evra racism row in 2011 and the bite on Chelsea's Branislav Ivanovic for which he is currently serving a 10-game ban.
But former team-mate Carragher, who is currently preparing for his new role as a pundit on Sky Sports, says the controversial forward should do what is best for his career if a deal is struck between the two clubs.
He said: "Luis Suarez can't stay at Liverpool forever just because they've supported him in certain situations.
"Liverpool were right to support him because he's their player but if he wasn't a very good player and wasn't producing on the pitch then Liverpool would sell him. Are they being loyal to a player if they sell him?
"I don't think Luis can be held to the club forever because they've stood by him in certain situations. You could argue that after the racism situation with Evra then he stayed last year because I'm sure he would have had offers last summer.
"I think loyalty in football, from both sides, is overplayed a bit. Basically Liverpool are loyal to Luis Suarez because he's such a good player and Luis has got to think of his own career and what he wants to achieve.
"I don't think that should really come into it or cloud people's judgments of what Luis should do."
Suarez signed a long-term deal with Liverpool last August and Carragher says the club hold all the cards when it comes to deciding his future.
He refused to criticise the forward for being tempted by another of the world's biggest clubs, but says he will only be allowed to leave Anfield if manager Brendan Rodgers and managing director Ian Ayre can strike the right deal.
"He's under contract," Carragher added. "He signed a contract 12 months ago, so if Liverpool really want to keep him then they can. I don't think it would affect the price if they sold him a year later.
"Liverpool were always going to be in this situation. Luis Suarez is playing for a team that's seventh in the Premier League and he's one of the best players in world football, so it would be pretty naive of us to feel that no club would be showing any interest in him.
"The manager, Ian Ayre and the owners knew other clubs would come calling for Luis Suarez and that's what's happened. If they look at a figure that's acceptable he can move on or if they want to keep Suarez because he's their best player then it's in their hands.
"As much as it's in Luis' court, it's in Liverpool's court to decide what they want to do; whether they want to keep the player or cash in on him."
Liverpool saw the form of another big-name striker, Fernando Torres, dip when he wanted to leave the club in 2010 but Carragher feels the Suarez situation is very different.
And he believes the club and its supporters would welcome him back, despite his comments, if he does not leave this summer.
"Fernando Torres, at times, wasn't giving his best on the pitch at Liverpool and that's why it was a mistake," Carragher added.
"I don't think that would happen with Luis Suarez ever. You know the type of player he is. Real Madrid is a massive club and they're in the Champions League but he loves Liverpool and he loves playing for Liverpool.
"But Liverpool can't just accept Luis Suarez going. If you want him you've got to pay top dollar for him so they will reinvest.
"I don't think he's said anything too bad. He got a bit carried away with his comments, but he hasn't said anything derogatory about Liverpool. He has said he's interested in going to Real Madrid - and we all know he'd think that privately, as any player would.
"I wouldn't get too carried away with the situation with Luis. He made some comments in Uruguay and was a bit naïve to think they wouldn't come back here, but he hasn't said anything too bad that can't be sorted out when he comes back here.
"The Liverpool supporters would love him to stay, of course they would."
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