The left-back, sidelined since the end of last season with a hip injury, is hoping an injection will help resolve the problem, but manager Brian McDermott said he may need an operation.
"Sam's got a hip problem," McDermott told LUTV. "He had an injection last week and we'll have a look at it over the next two or three days, but there's a possibility that he'll need an operation.
"We don't know yet, but that could put him out for a period of time. We'll know within the next week and it's a shame for him, but he's only 19 and if he has to be out for a while, we have to do the right thing by Sam.
"He's a good guy and a very, very good player. We'll wait and see."
Byram burst on to the first-team scene under previous manager Neil Warnock at the start of last season and went on to keep his place in the side throughout the campaign.
His outstanding performances did not go unnoticed outside Elland Road and he was linked with a B£10million move to Manchester City in April.