The Glaswegian giants will soon bid farewell to current No.1 Joe Hart, who announced last week that he will be retiring from football at the end of the season.
Shortly after the announcement, manager Brendan Rodgers admitted that there is "no doubt" the Hoops will be targeting at least one new goalkeeper in the summer transfer window.
"[The goalkeeping department is] something that has been earmarked with the club," Rodgers told Sky Sports. "So hopefully that will be something we can look at but we might need even more than one goalkeeper.
"Some of the guys that have been here for a while might want to go and play (backup goalkeepers Benjamin Siegrist and Scott Bain). They've been really loyal assistants to Joe, so we will have to see but there is no doubt that we will be in the market to bring in a goalkeeper."
Celtic have been persistently linked with a potential swoop for Liverpool's Kelleher, who starred between the sticks in last weekend's 1-0 EFL Cup final triumph over Chelsea at Wembley Stadium.
The Republic of Ireland international, who was allegedly the subject of a £15m January bid from Nottingham Forest, has had to play second fiddle behind Liverpool's first-choice Alisson Becker, starting just six Premier League games since the start of last season.
It is thought that Kelleher could be tempted by a move away from Anfield if he is guaranteed regular first-team football, although Football Insider reports that Celtic may find it difficult to convince the 26-year-old to move to Scotland amid interest from other Premier League clubs.
Celtic would also be required to stump up a 'big-money fee' in the region of £20m, a figure that may be out of the club's price range.
The Hoops could instead turn their attention to Trabzonspor star Cakir, as Turkish news outlet Gunebakis, via the Daily Record, claims that the 27-year-old is one of a number of names the Scottish Premiership champions are monitoring ahead of the summer transfer window.
Celtic are said to have sent scouts to watch Cakir in recent Trabzonspor matches - the 2021-22 Super Lig champions sit third in the Turkish top flight this season, although they are a whopping 26 points behind leaders Galatasaray.
Cakir has spent his entire career with Trabzonspor, making 232 appearances in all competitions since his debut in 2014, and he still has over three years remaining on his contract.
However, the 26-cap Turkey international is said to be unsettled at the Super Lig club and is seeking a fresh challenge away from his homeland.
Cakir was allegedly close to joining Southampton last year, but a transfer ultimately failed to materialise, while the likes of Liverpool and Borussia Dortmund have previously been linked with the shot-stopper.