It is even claimed by the paper's chief sports writer Jeremy Cross, that despite the high regard Byram is held in at the club, owner Massimo Cellino could be open to cashing in.
The article states: "Suspended Leeds president Massimo Cellino values Byram in the £10m bracket. But it is understood the controversial Italian is willing to reach a compromise with Reds bosses in order to strike a deal."
Would Byram fit in and what about Liverpool's current options?
The Star claim Liverpool are looking at Byram to fill a spot at right wing back, a position which could get the best out of him. He has previously been a regular for Leeds at right-back, but in 2015 has excelled in a position on the right wing.
Liverpool face losing Glen Johnson on a free transfer with his current deal expiring at the end of the season, while Javier Manquillo is one year into a two season long loan from Atletico Madrid. Jon Flanagan has missed the entire season with injury, and has not played at wing back for the club.
Do Leeds fans need to be worried?
The threat of losing their highly rated youngsters is likely to circulate all summer for Leeds, such is the high regard in which they are held.
Sam Byram is the most developed of any of the prospects to come through the academy, which makes his future even more important.
If Leeds sell him then it sends out a message that the rest of their top players are also for sale at the right price too, a situation they must avoid at all costs if they are to earn promotion next season.
What will Massimo Cellino do?
Massimo Cellino is currently suspended by the Football League and due to return next month. He was linked by The Mirror with selling the club to Red Bull just last week, and these stories only add to the uncertainty around the club.
Selling Byram would send out the wrong message about his intentions for the club, but if he wants a successful future at Leeds, he cannot ruin his relationship with the fanbase by cashing in on 'one of their own'.