Byram's contract expires at the end of this season, and this transfer window is the club's last opportunity to cash in on their own terms, rather than wait on a tribunal judged compensation package in the summer.
Here are three West Ham stars Leeds United should insist on signing as part of the negotiations...
Nikica Jelavic has struggled to hit the ground running since signing in the summer from Hull City, and he is yet to score for the Hammers. His only goal this season has come for Hull, but he does have 36 Premier League goals to his name in spells with Everton and the Tigers. With Enner Valencia and Diafra Sakho set to return to fitness and Jelavic already behind Andy Carroll, he faces being the fourth choice striker, and regular game time at Leeds should be more attractive.
If West Ham are worried about losing a right-back, then Joey O'Brien could help fill the void. The 29-year-old has not featured all season for the Hammers, having struggled early with an injury, and has not been selected since, and needs game time to help his Ireland ambitions for Euro 2016. The 29-year-old has experience and would be up upgrade on Scott Wootton, to compete with Gaetano Berardi.
Martin Samuelsen is currently on loan at Peterborough, but this agreement is soon to end unless an extension is agreed, and Leeds United should push to get the winger on their books until the end of the season. A summer signing from Manchester City's academy, the Norwegian is an outstanding prospect who scored a dazzling solo goal against Sheffield United at the weekend, and Leeds should try to work out a way of getting him in at Elland Road as part of any Byram deal.