It has been reported by the British tabloid that Liverpool have scouted the youngster in the last few games.
The Sun has also claimed that the 22-year-old, who can play as a right-back or as a right-winger, may be sold for around £5 million next month.
“The Anfield club know Byram’s representatives well, which could help them see off rivals when the market opens this week,” added the report.
“Leeds supremo Massimo Cellino has been looking for huge sums, but that may drop with Byram aiming to move when his deal ends in the summer.”
Byram has been on the books of Leeds his entire professional career so far and has a contract until the end of the season.
The Englishman is rated as one of the best young players in the Championship, but his form for the Whites so far this season has been rater mixed.
The youngster has scored two goals and created 18 chances in 18 Championship games this campaign.
Byram has a pass accuracy of 73% and a shot accuracy of 33%, and has won 30 headers, seven dribbles and 13 tackles.