Nuno Espirito Santo is believed to be prioritising the addition of a new shot-stopper before the winter transfer window closes at 11pm.
Matt Turner, a summer arrival from Arsenal, has featured regularly between the sticks for Forest this season, but the American has received criticism for a number of costly errors leading to goals.
Turner has conceded 10 goals in six appearances since the arrival of Nuno, while fellow summer addition Odysseas Vlachodimos has been restricted to just two FA Cup outings in the same period, conceding four goals.
Liverpool's backup goalkeeper Kelleher has emerged as a potential target for Forest, and the Daily Mail reports that they have submitted a late bid of £15m.
However, the report adds that the Reds have swiftly dismissed Forest's approach as they value the 25-year-old in excess of £20m.
Liverpool are said to have noted that Burnley signed 21-year-old James Trafford from Manchester City for £19m and they believe that Kelleher is at a more advanced stage of his development with greater potential, while he also boasts European and international experience.
Kelleher, an 11-cap Republic of Ireland international, joined Liverpool's academy in 2015 and worked his way through the club's youth ranks before making his competitive debut in September 2019.
However, the Cork-born shot-stopper has had to play second fiddle behind first-choice Alisson Becker, making just 33 first-team appearances in all competitions.
Kelleher has started just three Premier League games since the start of last season, including two this term against Fulham and Sheffield United in December.
Forest are also said to have attempted to recruit Crystal Palace's Sam Johnstone and Real Betis' Rui Silva in the final hours of deadline day, but they now appear to be closing on the signing of Strasbourg's Matz Sels.
Journalist Fabrizio Romano claims that Forest have agreed a deal to sign the 31-year-old Belgian for a fee between €6m (£5.1m) and €7m (£6m).