King, who played for Scotland Under 17s last month, turns 16 in January and can only sign a professional contract then. The Athletic have reported that Rangers are believed to have offered terms, but that Chelsea and Liverpool are both keeping tabs on the player.
The young centre half has already made several appearances for Graeme Murty's development squad this season.
It is thought that, should King decide to leave for England, a tribunal fee would sit in the region of £400,000-£500,000, similar to the figure Chelsea paid Rangers for Billy Gilmour.