Kelleher has established himself as the number 2 behind Alisson at Anfield and has impressed whenever handed opportunities by Jurgen Klopp.
The Republic of Ireland international was recently the hero as Liverpool won the EFL Cup, with Klopp keeping faith with him in the competition ahead of Alisson.
Kelleher though has still just enjoyed eight senior outings for Liverpool this season and he could be the subject of offers from elsewhere in the summer window.
Lawrenson does not think the 23-year-old should be in any rush to leave Anfield, as he can hone his skills in training against Mohamed Salah, Sadio Mane, Luis Diaz and Diogo Jota.
“No, why would you? [Kelleher should] stay at Liverpool and every day in training come up against Salah and Diaz and Mane and Jota”, Lawrenson said on Off The Ball.
“Absolutely not. The only thing he’s going to miss is if he went to a Championship team, maybe the battering he gets from crosses and corners and having to deal with that, but you can put that on in training.
“So no, stay where you are.”
Kelleher has a contract which Liverpool which is due to run until the summer of 2026.
