As speculation surrounding the Uruguayan strikers mooted summer move to Real Madrid continues, former teammate Carragher does not feel Suarez has to stay loyal to the Reds despite them supporting him through recent controversies.
"Luis Suarez can't stay at Liverpool forever just because they've supported him in certain situations," said Carragher told Sky Sports of Suarez who was involved in a bitter race row with Patrice Evra in 2011 and is currently serving a 10-match ban for biting Branislav Ivanovic.
"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."