Byram has become one of the most in-demand players in the Football League, due to his contract situation.
The 22-year-old has only six months left on his current deal at Elland Road, and Leeds chairman Massimo Cellino has already stated his desire to cash in on the youngster.
Leeds are said to be looking for £5 million for the defender, with both Everton and West Ham United keen to sign him.
Byram is more keen on a move to Goodison Park than Upton Park though, as he sees a quicker route into Roberto Martinez’s side’s first team.
With Seamus Coleman out with injury, Everton are lacking options at right-back and Byram could be ready to make the step up.
The Toffees have been keen to recruit Football League talent in recent times, and the rise of John Stones has encouraged Byram that a move to Merseyside would be right for him.