The Reds have been reportedly scouting Byram for several months and are now being tipped to make a bid at the end of the season, in the view of several national newspapers.
The Daily Star has even suggested than an £8m deal is close to being finalised, with Liverpool said to be confident of getting Byram signed up swiftly.
A host of top-flight clubs, including Sunderland and Southampton, are known to be admirers of the 21-year-old.
Liverpool are likely to be in the market for a right-back in the summer with high-earning Glen Johnson expected to move on and John Flanagan out of contract - and Javier Manquillo’s loan finishing at the end of the current campaign.
Byram is under contract at Leeds until the summer of 2016.