However, since the summer of 2013, Enrique has lost his place due to a succession of injuries, lack of form and competition for places, making just 12 league appearances and prompting questions over his future.
Many had expected Enrique to have moved on already, but the player is seemingly content to remain Anfield until his contract expires next summer - a decision which has riled Carragher.
“Enrique said he wanted to fight for his place. He added that he had looked to Spain for a move but economic conditions make it difficult,” Carragher explained in The Daily Mail.
“That translates as: I’m staying because no-one in Spain can afford me and I don’t care if I don’t play (just like last season) because I’m waiting to go on a free next summer when my contract expires.
“I am sure there are clubs in England or Spain who would want him for less money. It is about whether he really has the hunger to play even if it means a pay cut.”
With Joe Gomez arriving from Charlton Athletic and immediately making the first-team, it’s safe to say that manager Brendan Rodgers doesn’t have any plans for Enrique with Alberto Moreno also in contention.
It leaves the former Villarreal star with a decision to make: take a pay cut and play regularly elsewhere or sit around for another 12 months and reap the financial gains.
By sticking it out at Anfield, the 29-year-old is gambling on the idea that an established club will be willing to take him on next summer despite having barely featured for three years.