Having played second fiddle to Alisson Becker at Anfield for a number of years, Kelleher could be on the way out this summer in search of regular game time elsewhere.
The Republic of Ireland international still has three years left on his contract with Liverpool, but he has made just three senior appearances in all tournaments during the 2022-23 campaign.
Furthermore, all three of those starts have come in cup competitions, including just one outing from Liverpool's three FA Cup matches in their third-round replay at Wolverhampton Wanderers.
Kelleher unsurprisingly began Liverpool's EFL Cup defence in goal, but their run came to a swift end in the fourth round with defeat to Manchester City, and he has been unable to get a look-in over Alisson in the Premier League or Champions League.
As a result, the 24-year-old is being tipped to end his eight-year stay on Merseyside, and both Brighton and Brentford are believed to have been alerted to his availability.
However, the Irish Independent claims that Tottenham are also ready to test Liverpool's resolve over Kelleher, who is aware that Alisson will likely remain Liverpool's number one for years to come.
The report claims that Liverpool are willing to sell Kelleher for the right price this summer, but they are prepared to demand as much as £20m to sanction a transfer for their number two goalkeeper.
Hugo Lloris has been Tottenham's undisputed first-choice goalkeeper for over a decade, but the former France international will turn 37 this year and is only under contract with the club until 2024.
Lloris has also shown a propensity for mistakes this season, making four errors directly leading to a goal in all competitions, the joint-most for Premier League shot-stoppers alongside Manchester United's David de Gea.
The former Lyon man may have already played his final game in a Tottenham shirt due to a season-ending hip injury, allowing summer signing Fraser Forster to deputise in between the sticks for the final few weeks of the campaign.
However, at 35 years of age, Forster - whose own contract expires at the end of next season too - is not seen as a long-term solution to Tottenham's goalkeeping dilemma, while academy graduates Brandon Austin and Alfie Whiteman remain out of favour.
Kelleher departed Irish outfit Ringmahon Rangers in 2015 to join Liverpool's academy, but he has made just 20 senior appearances for the Reds since rising through the youth ranks, keeping 10 clean sheets.
The 24-year-old played an integral role in Liverpool's EFL Cup triumph last season, starting in the showpiece match against Chelsea and scoring his side's 11th and final penalty.
As well as the EFL Cup, Kelleher has been a part of winning squads in the Champions League, Premier League, FA Cup and UEFA Super Cup, but he has made just four appearances in the English top flight to date.