The Hoops are on the lookout for a new number one goalkeeper following the retirement of Joe Hart, who has now hung up his gloves after three years at Parkhead.
According to Football Insider, Celtic are willing to fork out as much as £10m for the 25-year-old, but Liverpool are set to hold out for a larger sum.
Kelleher has had to play second fiddle to first-choice shot-stopper Alisson Becker at Liverpool and has subsequently been reduced to 47 appearances in all competitions since and including his debut in 2019, although 26 of those were made last season.
The Republic of Ireland international still has two years remaining on his contract at Anfield, but he has hinted at a summer move after confirming his desire to become a number one goalkeeper.
"I've had such an amazing time at Liverpool. It's been a crazy journey for me and one that I've absolutely loved," Kelleher told The Athletic. "I love the club, I love the fans and I have a great relationship with the players and the staff. Whether it's here at Liverpool or somewhere else, I do feel the next step for me is to be a No.1."
Kelleher is not the only goalkeeper said to be on Celtic's radar, as the Glaswegian giants have also been credited with an interest in Manchester United's number two Altay Bayindir.
Staying in Scotland and Rangers are reportedly interested in signing 19-year-old Arsenal defender Reuell Walters.
The Gunners academy graduate is yet to make a senior appearance for the club and is facing an uncertain future at the Emirates Stadium after recently rejecting a contract extension.
Walters, who made 23 youth team appearances last season, is due to see his current deal expire at the end of this month, although Arsenal have confirmed that they remain in discussions with the youngster over a possible renewal.
Should Walters decide to seek pastures new, Rangers could look to step up their interest in the right-back, according to The Scotsman.
Philippe Clement is overseeing a major squad overhaul at Ibrox this summer and has already seen five first-team players depart, while left-back Jefte and winger Oscar Cortes have both joined the club, the latter has extended his loan deal that includes an obligation to buy in 2025.
Elsewhere, Aston Villa are reportedly on the verge of completing the signing of Luton Town midfielder Ross Barkley.
The 30-year-old was snapped up by the Hatters on a free transfer last summer and he excelled for the club in their first-ever season as a Premier League club, chipping in with five goals and six assists in 37 appearances.
However, following Luton's relegation from the top flight, Barkley now appears set to swap the Championship for the Champions League, as multiple reports, including one from talkSPORT, claim that Aston Villa have agreed a £5m deal to recruit the midfielder.
The opportunity to continue in the Premier League and compete in Europe is seemingly too good to turn down for Barkley, who recently stated that he can compete in his 'prime' for the 'next three years'.
Barkley is already familiar with Villa Park as he played for the Lions on loan from Chelsea during the 2020-21 season, making 24 appearances in all competitions.