The former centre-back won eleven trophies as a player with his boyhood club, and was clear that he thought Rodgers should be looking to add a trophy by the end of this season, his third at the club.
Liverpool take on Aston Villa on Sunday in the FA Cup semi finals having only recently dispatched Blackburn in a quarter final replay, with a meeting against Arsenal or Reading waiting if they are successful.
“Brendan Rodgers will have done three years in the job and I don’t think a Liverpool manager has gone three years without winning a trophy," said Carragher.
“When you play for a side like Liverpool, Manchester United or Arsenal, or as a manager, then you have to win trophies."
The former Liverpool star said that claiming trophies is ultimately the aim of players and managers, with the prospect of finishing fourth in the league only important for a club's board and how they can plan for the future.
The former England defender was in no doubt as to what players remember when they retire, and was honest with what would be most important to him were he still playing.
“Top four is good for the people above," he added. "For money, more players, but at the end of your time you won’t be judged on how many times you get in the Champions League - it’s how many trophies you’ve won.
“If it’s one or another as a player I’d always pick winning the FA Cup.”