Byram’s current deal is set to expire at the end of the current campaign and he has already turned down a number of offers to pen a new contract.
Leeds owner Massimo Cellino revealed he was “deeply offended” after Byram turned down the chance to extend his stay.
According to the Sunday Mirror West Ham, Crystal Palace, Leicester and Norwich are all keen on acquiring the 22-year-old in January.
It is understood that Leeds will try to hold out for a fee in the region of £8million as they believe he has a big future.
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.