According to The Sun, The Reds are keen to keep hold of the Ireland international who helped Jurgen Klopp's side secure the Carabao Cup by converting the winning spot-kick against Chelsea last month.
Klopp branded Kelleher the 'world's best No.2' in an interview after his team's triumph at Wembley as he held his nerve.
"Alisson Becker is the best goalie in the world for me. There are goalies out there but this goalie is absolutely insane. But to be 100 per cent honest, for me Caoimhin Kelleher is the best No.2 in the world as well, especially for the way we play."
However, the shot-stopper is unlikely to earn international honours on a regular basis any time soon due to his role at Anfield.
Republic of Ireland boss Stephen Kenny has confirmed that Manchester City goalkeeper Gavin Bazunu still has the edge over Kelleher as he stars on loan at Portsmouth.
The Irish are set to face Belgium and Lithuania later this month in friendly matches, but Bazunu, 20, is set to wear the number one shirt.
Shay Given has previously claimed Kelleher should seek a move away in search of regular game time to boost his chances on the international stage.
"He's in a bit of a predicament Kelleher", Given told Premier Sports.
"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.
"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 20-years old and has played 50 club games, Kelleher has played around ten…
"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."
Having been with Liverpool since 2015, Kelleher made his first-team debut in 2019, and has made two appearances in the Premier League this season.
Liverpool are keen to keep hold of the 23-year-old with German Loris Karius and experienced Spaniard Adrian set to leave the club in the summer.
Cork-born Kelleher hasn't put a foot wrong whenever called upon by Klopp, although it is difficult to see him getting consistent playing time at Anfield in the short-term considering the calibre of Alisson.