The report states they are one of four teams interested in the soon to be out of contract 22-year-old, along with Crystal Palace, West Ham, and Norwich City.
Byram was nominated for December's Championship Player of the Month Award after netting three goals for Leeds, playing in a more advanced role.
He is still weighing up whether to sign a new deal or move elsewhere, but Leicester's interest specifically, could prove to be decisive.
Leicester have stunned everybody with their start to the 2015/16 season, and currently sit third in the Premier League.
Amid a weak field in the top flight this campaign, Claudio Ranieri's team are in with a real chance of finishing the season in the top four, and spending in the transfer window this going to be essential to them kicking on between now and May.
The team has an opportunity to qualify for a Champions League place next season, and that is an exciting challenge for any player to be a part of.
For Sam Byram it would potentially be more attractive than a long shot attempt to reach the Championship play-offs with Leeds.
West Ham, Norwich and Crystal Palace all have targets to play for too, but it is Leicester who have the biggest prize to play for, and that should be the only reason Byram leaves Leeds.