Kelleher has cemented his position as Liverpool's back-up keeper to first choice Alisson Becker, with his performances in the Carabao Cup catching the eye this season.
Yet Given suggests the Cork-born stopper, who started for Liverpool in their FA Cup win against Shrewsbury on Sunday, needs to play more regularly at club level as he looks to establish himself as Ireland's No.1 ahead of Gavin Bazunu.
"He's in a bit of a predicament," said Given of Kelleher. "He's 23-years old, I'm trying to bring myself back to when I was that age and I'm not sure how many games I would have played, but I think he has only played ten or 11 games.
"He's not a kid anymore, 23 is an age where I think he should be playing regular first team football.
"The biggest problem he has is that Alisson is ahead of him. Alisson is in the prime of his life and could be for the next five or six years. Does Kelleher sit there and be happy as a number two? They've got Adrian who can sit on the bench and back (Alisson) up.
"For me, it's so important to play games. He has been training there for a few years and signed a new five-year contract last summer, so he is part of the club's long-term future.
"Even from a Liverpool point of view, they need to get him out playing, get him that experience, then bring him back and get him challenging Alisson."
Given went on to suggest Kelleher should be pushing Liverpool to allow him to leave in a loan deal in this month's transfer window, as he added: "If it was me, I'd be knocking Klopp's door down, I really would. I'd be saying that I need to go out on loan, go and play.
"January is a time to do that, I'd be on to my agent and manager to get me playing games. It's not just at Liverpool, he's got Bazunu at Ireland as well. He is only 19-years old and has played 50 club games, Kelleher has played around ten (actual number is 13 Liverpool appearances).
"I think for all parties concerned the best thing would be to go on loan. I think he's good enough to play in the first team, he's a really good goalkeeper. I think he wants to show people that."