The Toffees are understood to be monitoring the 21-year-old with ongoing speculation concerning the future of Republic of Ireland international Seamus Coleman, who is wanted by Manchester United.
Everton have insisted that neither Coleman or John Stones are for sale, with boss Roberto Martinez keen to keep his squad together, but the rumours regarding the pair have failed to cease.
The Red Devils are considering a move for Coleman, 26, with the Merseysiders looking at possible alternatives if they are tabled an offer they can’t refuse.
Byram has a year left on his Elland Road deal, with Sunderland also interested in him, with the likes of Newcastle United and Aston Villa also credited with an interest.
In a recent interview, Byram spoke about looking forward to a new season with Leeds, but he has made no secret on his longer-term desire to test himself in the top-flight.
Sunderland have a firm interest in Byram and could make a move if they fail to recruit loan right-back Karl Jenkinson from Arsenal - with the Wearsiders having agreed a loan fee of £2.5m with the Gunners.
Jenkinson is also wanted by West Ham, where he spent a productive loan spell last term with the Hammers being the favourites to land him.
Leeds are battling out it out with Wolves to land Leicester City striker Chris Wood, with both clubs having seen bids of around £2m accepted by the Foxes.
Wigan Athletic and Sheffield United are both keen on landing United forward Billy Sharp, with a Championship side also believed to be interested.