Kelleher, 27, joined Brentford on a permanent basis last summer after deciding that first-choice football mattered more than the comfort of a back-up role at Anfield.
He has since established himself as one of the most consistent goalkeepers in the Premier League, keeping nine clean sheets in 32 league appearances.
Speaking to The Athletic, he described what the two coaches have in common.
"They are both very, very passionate and very good with people. They get to know you as a person first, before the footballer."
He has found life in west London rewarding but has not pretended the move was emotionally simple.
"Liverpool is a special place for me. I lived in the city centre for five years and loved it there. I miss the people at the club. I have many good friends there."
He also expressed a warmth for the city and its people that surprised him when he first arrived.
"I miss them, and I miss the city and the Scousers. I think they are like the Irish. They are quite funny. I miss them."
Kelleher made his senior debut for Liverpool under Klopp in autumn 2019 and earned a growing reputation as one of Europe's most capable backup goalkeepers before his move. He is contracted at Brentford until 2030 and has 31 senior caps for the Republic of Ireland.